src/HOL/Library/Formal_Power_Series.thy
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```     1 (*  Title:      HOL/Library/Formal_Power_Series.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 header{* A formalization of formal power series *}
```
```     6
```
```     7 theory Formal_Power_Series
```
```     8 imports Complex_Main Binomial
```
```     9 begin
```
```    10
```
```    11
```
```    12 subsection {* The type of formal power series*}
```
```    13
```
```    14 typedef (open) 'a fps = "{f :: nat \<Rightarrow> 'a. True}"
```
```    15   morphisms fps_nth Abs_fps
```
```    16   by simp
```
```    17
```
```    18 notation fps_nth (infixl "\$" 75)
```
```    19
```
```    20 lemma expand_fps_eq: "p = q \<longleftrightarrow> (\<forall>n. p \$ n = q \$ n)"
```
```    21   by (simp add: fps_nth_inject [symmetric] fun_eq_iff)
```
```    22
```
```    23 lemma fps_ext: "(\<And>n. p \$ n = q \$ n) \<Longrightarrow> p = q"
```
```    24   by (simp add: expand_fps_eq)
```
```    25
```
```    26 lemma fps_nth_Abs_fps [simp]: "Abs_fps f \$ n = f n"
```
```    27   by (simp add: Abs_fps_inverse)
```
```    28
```
```    29 text{* Definition of the basic elements 0 and 1 and the basic operations of addition, negation and multiplication *}
```
```    30
```
```    31 instantiation fps :: (zero) zero
```
```    32 begin
```
```    33
```
```    34 definition fps_zero_def:
```
```    35   "0 = Abs_fps (\<lambda>n. 0)"
```
```    36
```
```    37 instance ..
```
```    38 end
```
```    39
```
```    40 lemma fps_zero_nth [simp]: "0 \$ n = 0"
```
```    41   unfolding fps_zero_def by simp
```
```    42
```
```    43 instantiation fps :: ("{one, zero}") one
```
```    44 begin
```
```    45
```
```    46 definition fps_one_def:
```
```    47   "1 = Abs_fps (\<lambda>n. if n = 0 then 1 else 0)"
```
```    48
```
```    49 instance ..
```
```    50 end
```
```    51
```
```    52 lemma fps_one_nth [simp]: "1 \$ n = (if n = 0 then 1 else 0)"
```
```    53   unfolding fps_one_def by simp
```
```    54
```
```    55 instantiation fps :: (plus)  plus
```
```    56 begin
```
```    57
```
```    58 definition fps_plus_def:
```
```    59   "op + = (\<lambda>f g. Abs_fps (\<lambda>n. f \$ n + g \$ n))"
```
```    60
```
```    61 instance ..
```
```    62 end
```
```    63
```
```    64 lemma fps_add_nth [simp]: "(f + g) \$ n = f \$ n + g \$ n"
```
```    65   unfolding fps_plus_def by simp
```
```    66
```
```    67 instantiation fps :: (minus) minus
```
```    68 begin
```
```    69
```
```    70 definition fps_minus_def:
```
```    71   "op - = (\<lambda>f g. Abs_fps (\<lambda>n. f \$ n - g \$ n))"
```
```    72
```
```    73 instance ..
```
```    74 end
```
```    75
```
```    76 lemma fps_sub_nth [simp]: "(f - g) \$ n = f \$ n - g \$ n"
```
```    77   unfolding fps_minus_def by simp
```
```    78
```
```    79 instantiation fps :: (uminus) uminus
```
```    80 begin
```
```    81
```
```    82 definition fps_uminus_def:
```
```    83   "uminus = (\<lambda>f. Abs_fps (\<lambda>n. - (f \$ n)))"
```
```    84
```
```    85 instance ..
```
```    86 end
```
```    87
```
```    88 lemma fps_neg_nth [simp]: "(- f) \$ n = - (f \$ n)"
```
```    89   unfolding fps_uminus_def by simp
```
```    90
```
```    91 instantiation fps :: ("{comm_monoid_add, times}")  times
```
```    92 begin
```
```    93
```
```    94 definition fps_times_def:
```
```    95   "op * = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f \$ i * g \$ (n - i)))"
```
```    96
```
```    97 instance ..
```
```    98 end
```
```    99
```
```   100 lemma fps_mult_nth: "(f * g) \$ n = (\<Sum>i=0..n. f\$i * g\$(n - i))"
```
```   101   unfolding fps_times_def by simp
```
```   102
```
```   103 declare atLeastAtMost_iff[presburger]
```
```   104 declare Bex_def[presburger]
```
```   105 declare Ball_def[presburger]
```
```   106
```
```   107 lemma mult_delta_left:
```
```   108   fixes x y :: "'a::mult_zero"
```
```   109   shows "(if b then x else 0) * y = (if b then x * y else 0)"
```
```   110   by simp
```
```   111
```
```   112 lemma mult_delta_right:
```
```   113   fixes x y :: "'a::mult_zero"
```
```   114   shows "x * (if b then y else 0) = (if b then x * y else 0)"
```
```   115   by simp
```
```   116
```
```   117 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
```
```   118   by auto
```
```   119 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
```
```   120   by auto
```
```   121
```
```   122 subsection{* Formal power series form a commutative ring with unity, if the range of sequences
```
```   123   they represent is a commutative ring with unity*}
```
```   124
```
```   125 instance fps :: (semigroup_add) semigroup_add
```
```   126 proof
```
```   127   fix a b c :: "'a fps" show "a + b + c = a + (b + c)"
```
```   128     by (simp add: fps_ext add_assoc)
```
```   129 qed
```
```   130
```
```   131 instance fps :: (ab_semigroup_add) ab_semigroup_add
```
```   132 proof
```
```   133   fix a b :: "'a fps" show "a + b = b + a"
```
```   134     by (simp add: fps_ext add_commute)
```
```   135 qed
```
```   136
```
```   137 lemma fps_mult_assoc_lemma:
```
```   138   fixes k :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
```
```   139   shows "(\<Sum>j=0..k. \<Sum>i=0..j. f i (j - i) (n - j)) =
```
```   140          (\<Sum>j=0..k. \<Sum>i=0..k - j. f j i (n - j - i))"
```
```   141 proof (induct k)
```
```   142   case 0 show ?case by simp
```
```   143 next
```
```   144   case (Suc k) thus ?case
```
```   145     by (simp add: Suc_diff_le setsum_addf add_assoc
```
```   146              cong: strong_setsum_cong)
```
```   147 qed
```
```   148
```
```   149 instance fps :: (semiring_0) semigroup_mult
```
```   150 proof
```
```   151   fix a b c :: "'a fps"
```
```   152   show "(a * b) * c = a * (b * c)"
```
```   153   proof (rule fps_ext)
```
```   154     fix n :: nat
```
```   155     have "(\<Sum>j=0..n. \<Sum>i=0..j. a\$i * b\$(j - i) * c\$(n - j)) =
```
```   156           (\<Sum>j=0..n. \<Sum>i=0..n - j. a\$j * b\$i * c\$(n - j - i))"
```
```   157       by (rule fps_mult_assoc_lemma)
```
```   158     thus "((a * b) * c) \$ n = (a * (b * c)) \$ n"
```
```   159       by (simp add: fps_mult_nth setsum_right_distrib
```
```   160                     setsum_left_distrib mult_assoc)
```
```   161   qed
```
```   162 qed
```
```   163
```
```   164 lemma fps_mult_commute_lemma:
```
```   165   fixes n :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
```
```   166   shows "(\<Sum>i=0..n. f i (n - i)) = (\<Sum>i=0..n. f (n - i) i)"
```
```   167 proof (rule setsum_reindex_cong)
```
```   168   show "inj_on (\<lambda>i. n - i) {0..n}"
```
```   169     by (rule inj_onI) simp
```
```   170   show "{0..n} = (\<lambda>i. n - i) ` {0..n}"
```
```   171     by (auto, rule_tac x="n - x" in image_eqI, simp_all)
```
```   172 next
```
```   173   fix i assume "i \<in> {0..n}"
```
```   174   hence "n - (n - i) = i" by simp
```
```   175   thus "f (n - i) i = f (n - i) (n - (n - i))" by simp
```
```   176 qed
```
```   177
```
```   178 instance fps :: (comm_semiring_0) ab_semigroup_mult
```
```   179 proof
```
```   180   fix a b :: "'a fps"
```
```   181   show "a * b = b * a"
```
```   182   proof (rule fps_ext)
```
```   183     fix n :: nat
```
```   184     have "(\<Sum>i=0..n. a\$i * b\$(n - i)) = (\<Sum>i=0..n. a\$(n - i) * b\$i)"
```
```   185       by (rule fps_mult_commute_lemma)
```
```   186     thus "(a * b) \$ n = (b * a) \$ n"
```
```   187       by (simp add: fps_mult_nth mult_commute)
```
```   188   qed
```
```   189 qed
```
```   190
```
```   191 instance fps :: (monoid_add) monoid_add
```
```   192 proof
```
```   193   fix a :: "'a fps" show "0 + a = a "
```
```   194     by (simp add: fps_ext)
```
```   195 next
```
```   196   fix a :: "'a fps" show "a + 0 = a "
```
```   197     by (simp add: fps_ext)
```
```   198 qed
```
```   199
```
```   200 instance fps :: (comm_monoid_add) comm_monoid_add
```
```   201 proof
```
```   202   fix a :: "'a fps" show "0 + a = a "
```
```   203     by (simp add: fps_ext)
```
```   204 qed
```
```   205
```
```   206 instance fps :: (semiring_1) monoid_mult
```
```   207 proof
```
```   208   fix a :: "'a fps" show "1 * a = a"
```
```   209     by (simp add: fps_ext fps_mult_nth mult_delta_left setsum_delta)
```
```   210 next
```
```   211   fix a :: "'a fps" show "a * 1 = a"
```
```   212     by (simp add: fps_ext fps_mult_nth mult_delta_right setsum_delta')
```
```   213 qed
```
```   214
```
```   215 instance fps :: (cancel_semigroup_add) cancel_semigroup_add
```
```   216 proof
```
```   217   fix a b c :: "'a fps"
```
```   218   assume "a + b = a + c" then show "b = c"
```
```   219     by (simp add: expand_fps_eq)
```
```   220 next
```
```   221   fix a b c :: "'a fps"
```
```   222   assume "b + a = c + a" then show "b = c"
```
```   223     by (simp add: expand_fps_eq)
```
```   224 qed
```
```   225
```
```   226 instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
```
```   227 proof
```
```   228   fix a b c :: "'a fps"
```
```   229   assume "a + b = a + c" then show "b = c"
```
```   230     by (simp add: expand_fps_eq)
```
```   231 qed
```
```   232
```
```   233 instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..
```
```   234
```
```   235 instance fps :: (group_add) group_add
```
```   236 proof
```
```   237   fix a :: "'a fps" show "- a + a = 0"
```
```   238     by (simp add: fps_ext)
```
```   239 next
```
```   240   fix a b :: "'a fps" show "a - b = a + - b"
```
```   241     by (simp add: fps_ext diff_minus)
```
```   242 qed
```
```   243
```
```   244 instance fps :: (ab_group_add) ab_group_add
```
```   245 proof
```
```   246   fix a :: "'a fps"
```
```   247   show "- a + a = 0"
```
```   248     by (simp add: fps_ext)
```
```   249 next
```
```   250   fix a b :: "'a fps"
```
```   251   show "a - b = a + - b"
```
```   252     by (simp add: fps_ext)
```
```   253 qed
```
```   254
```
```   255 instance fps :: (zero_neq_one) zero_neq_one
```
```   256   by default (simp add: expand_fps_eq)
```
```   257
```
```   258 instance fps :: (semiring_0) semiring
```
```   259 proof
```
```   260   fix a b c :: "'a fps"
```
```   261   show "(a + b) * c = a * c + b * c"
```
```   262     by (simp add: expand_fps_eq fps_mult_nth left_distrib setsum_addf)
```
```   263 next
```
```   264   fix a b c :: "'a fps"
```
```   265   show "a * (b + c) = a * b + a * c"
```
```   266     by (simp add: expand_fps_eq fps_mult_nth right_distrib setsum_addf)
```
```   267 qed
```
```   268
```
```   269 instance fps :: (semiring_0) semiring_0
```
```   270 proof
```
```   271   fix a:: "'a fps" show "0 * a = 0"
```
```   272     by (simp add: fps_ext fps_mult_nth)
```
```   273 next
```
```   274   fix a:: "'a fps" show "a * 0 = 0"
```
```   275     by (simp add: fps_ext fps_mult_nth)
```
```   276 qed
```
```   277
```
```   278 instance fps :: (semiring_0_cancel) semiring_0_cancel ..
```
```   279
```
```   280 subsection {* Selection of the nth power of the implicit variable in the infinite sum*}
```
```   281
```
```   282 lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f \$n \<noteq> 0)"
```
```   283   by (simp add: expand_fps_eq)
```
```   284
```
```   285 lemma fps_nonzero_nth_minimal:
```
```   286   "f \<noteq> 0 \<longleftrightarrow> (\<exists>n. f \$ n \<noteq> 0 \<and> (\<forall>m<n. f \$ m = 0))"
```
```   287 proof
```
```   288   let ?n = "LEAST n. f \$ n \<noteq> 0"
```
```   289   assume "f \<noteq> 0"
```
```   290   then have "\<exists>n. f \$ n \<noteq> 0"
```
```   291     by (simp add: fps_nonzero_nth)
```
```   292   then have "f \$ ?n \<noteq> 0"
```
```   293     by (rule LeastI_ex)
```
```   294   moreover have "\<forall>m<?n. f \$ m = 0"
```
```   295     by (auto dest: not_less_Least)
```
```   296   ultimately have "f \$ ?n \<noteq> 0 \<and> (\<forall>m<?n. f \$ m = 0)" ..
```
```   297   then show "\<exists>n. f \$ n \<noteq> 0 \<and> (\<forall>m<n. f \$ m = 0)" ..
```
```   298 next
```
```   299   assume "\<exists>n. f \$ n \<noteq> 0 \<and> (\<forall>m<n. f \$ m = 0)"
```
```   300   then show "f \<noteq> 0" by (auto simp add: expand_fps_eq)
```
```   301 qed
```
```   302
```
```   303 lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f \$ n = g \$n)"
```
```   304   by (rule expand_fps_eq)
```
```   305
```
```   306 lemma fps_setsum_nth: "(setsum f S) \$ n = setsum (\<lambda>k. (f k) \$ n) S"
```
```   307 proof (cases "finite S")
```
```   308   assume "\<not> finite S" then show ?thesis by simp
```
```   309 next
```
```   310   assume "finite S"
```
```   311   then show ?thesis by (induct set: finite) auto
```
```   312 qed
```
```   313
```
```   314 subsection{* Injection of the basic ring elements and multiplication by scalars *}
```
```   315
```
```   316 definition
```
```   317   "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)"
```
```   318
```
```   319 lemma fps_nth_fps_const [simp]: "fps_const c \$ n = (if n = 0 then c else 0)"
```
```   320   unfolding fps_const_def by simp
```
```   321
```
```   322 lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0"
```
```   323   by (simp add: fps_ext)
```
```   324
```
```   325 lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1"
```
```   326   by (simp add: fps_ext)
```
```   327
```
```   328 lemma fps_const_neg [simp]: "- (fps_const (c::'a::ring)) = fps_const (- c)"
```
```   329   by (simp add: fps_ext)
```
```   330
```
```   331 lemma fps_const_add [simp]: "fps_const (c::'a\<Colon>monoid_add) + fps_const d = fps_const (c + d)"
```
```   332   by (simp add: fps_ext)
```
```   333 lemma fps_const_sub [simp]: "fps_const (c::'a\<Colon>group_add) - fps_const d = fps_const (c - d)"
```
```   334   by (simp add: fps_ext)
```
```   335 lemma fps_const_mult[simp]: "fps_const (c::'a\<Colon>ring) * fps_const d = fps_const (c * d)"
```
```   336   by (simp add: fps_eq_iff fps_mult_nth setsum_0')
```
```   337
```
```   338 lemma fps_const_add_left: "fps_const (c::'a\<Colon>monoid_add) + f = Abs_fps (\<lambda>n. if n = 0 then c + f\$0 else f\$n)"
```
```   339   by (simp add: fps_ext)
```
```   340
```
```   341 lemma fps_const_add_right: "f + fps_const (c::'a\<Colon>monoid_add) = Abs_fps (\<lambda>n. if n = 0 then f\$0 + c else f\$n)"
```
```   342   by (simp add: fps_ext)
```
```   343
```
```   344 lemma fps_const_mult_left: "fps_const (c::'a\<Colon>semiring_0) * f = Abs_fps (\<lambda>n. c * f\$n)"
```
```   345   unfolding fps_eq_iff fps_mult_nth
```
```   346   by (simp add: fps_const_def mult_delta_left setsum_delta)
```
```   347
```
```   348 lemma fps_const_mult_right: "f * fps_const (c::'a\<Colon>semiring_0) = Abs_fps (\<lambda>n. f\$n * c)"
```
```   349   unfolding fps_eq_iff fps_mult_nth
```
```   350   by (simp add: fps_const_def mult_delta_right setsum_delta')
```
```   351
```
```   352 lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)\$n = c* f\$n"
```
```   353   by (simp add: fps_mult_nth mult_delta_left setsum_delta)
```
```   354
```
```   355 lemma fps_mult_right_const_nth [simp]: "(f * fps_const (c::'a::semiring_1))\$n = f\$n * c"
```
```   356   by (simp add: fps_mult_nth mult_delta_right setsum_delta')
```
```   357
```
```   358 subsection {* Formal power series form an integral domain*}
```
```   359
```
```   360 instance fps :: (ring) ring ..
```
```   361
```
```   362 instance fps :: (ring_1) ring_1
```
```   363   by (intro_classes, auto simp add: diff_minus left_distrib)
```
```   364
```
```   365 instance fps :: (comm_ring_1) comm_ring_1
```
```   366   by (intro_classes, auto simp add: diff_minus left_distrib)
```
```   367
```
```   368 instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors
```
```   369 proof
```
```   370   fix a b :: "'a fps"
```
```   371   assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0"
```
```   372   then obtain i j where i: "a\$i\<noteq>0" "\<forall>k<i. a\$k=0"
```
```   373     and j: "b\$j \<noteq>0" "\<forall>k<j. b\$k =0" unfolding fps_nonzero_nth_minimal
```
```   374     by blast+
```
```   375   have "(a * b) \$ (i+j) = (\<Sum>k=0..i+j. a\$k * b\$(i+j-k))"
```
```   376     by (rule fps_mult_nth)
```
```   377   also have "\<dots> = (a\$i * b\$(i+j-i)) + (\<Sum>k\<in>{0..i+j}-{i}. a\$k * b\$(i+j-k))"
```
```   378     by (rule setsum_diff1') simp_all
```
```   379   also have "(\<Sum>k\<in>{0..i+j}-{i}. a\$k * b\$(i+j-k)) = 0"
```
```   380     proof (rule setsum_0' [rule_format])
```
```   381       fix k assume "k \<in> {0..i+j} - {i}"
```
```   382       then have "k < i \<or> i+j-k < j" by auto
```
```   383       then show "a\$k * b\$(i+j-k) = 0" using i j by auto
```
```   384     qed
```
```   385   also have "a\$i * b\$(i+j-i) + 0 = a\$i * b\$j" by simp
```
```   386   also have "a\$i * b\$j \<noteq> 0" using i j by simp
```
```   387   finally have "(a*b) \$ (i+j) \<noteq> 0" .
```
```   388   then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast
```
```   389 qed
```
```   390
```
```   391 instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors ..
```
```   392
```
```   393 instance fps :: (idom) idom ..
```
```   394
```
```   395 instantiation fps :: (comm_ring_1) number_ring
```
```   396 begin
```
```   397 definition number_of_fps_def: "(number_of k::'a fps) = of_int k"
```
```   398
```
```   399 instance proof
```
```   400 qed (rule number_of_fps_def)
```
```   401 end
```
```   402
```
```   403 lemma number_of_fps_const: "(number_of k::('a::comm_ring_1) fps) = fps_const (of_int k)"
```
```   404
```
```   405 proof(induct k rule: int_induct [where k=0])
```
```   406   case base thus ?case unfolding number_of_fps_def of_int_0 by simp
```
```   407 next
```
```   408   case (step1 i) thus ?case unfolding number_of_fps_def
```
```   409     by (simp add: fps_const_add[symmetric] del: fps_const_add)
```
```   410 next
```
```   411   case (step2 i) thus ?case unfolding number_of_fps_def
```
```   412     by (simp add: fps_const_sub[symmetric] del: fps_const_sub)
```
```   413 qed
```
```   414 subsection{* The eXtractor series X*}
```
```   415
```
```   416 lemma minus_one_power_iff: "(- (1::'a :: {comm_ring_1})) ^ n = (if even n then 1 else - 1)"
```
```   417   by (induct n, auto)
```
```   418
```
```   419 definition "X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)"
```
```   420 lemma X_mult_nth[simp]: "(X * (f :: ('a::semiring_1) fps)) \$n = (if n = 0 then 0 else f \$ (n - 1))"
```
```   421 proof-
```
```   422   {assume n: "n \<noteq> 0"
```
```   423     have fN: "finite {0 .. n}" by simp
```
```   424     have "(X * f) \$n = (\<Sum>i = 0..n. X \$ i * f \$ (n - i))" by (simp add: fps_mult_nth)
```
```   425     also have "\<dots> = f \$ (n - 1)"
```
```   426       using n by (simp add: X_def mult_delta_left setsum_delta [OF fN])
```
```   427   finally have ?thesis using n by simp }
```
```   428   moreover
```
```   429   {assume n: "n=0" hence ?thesis by (simp add: fps_mult_nth X_def)}
```
```   430   ultimately show ?thesis by blast
```
```   431 qed
```
```   432
```
```   433 lemma X_mult_right_nth[simp]: "((f :: ('a::comm_semiring_1) fps) * X) \$n = (if n = 0 then 0 else f \$ (n - 1))"
```
```   434   by (metis X_mult_nth mult_commute)
```
```   435
```
```   436 lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then (1::'a::comm_ring_1) else 0)"
```
```   437 proof(induct k)
```
```   438   case 0 thus ?case by (simp add: X_def fps_eq_iff)
```
```   439 next
```
```   440   case (Suc k)
```
```   441   {fix m
```
```   442     have "(X^Suc k) \$ m = (if m = 0 then (0::'a) else (X^k) \$ (m - 1))"
```
```   443       by (simp add: power_Suc del: One_nat_def)
```
```   444     then     have "(X^Suc k) \$ m = (if m = Suc k then (1::'a) else 0)"
```
```   445       using Suc.hyps by (auto cong del: if_weak_cong)}
```
```   446   then show ?case by (simp add: fps_eq_iff)
```
```   447 qed
```
```   448
```
```   449 lemma X_power_mult_nth: "(X^k * (f :: ('a::comm_ring_1) fps)) \$n = (if n < k then 0 else f \$ (n - k))"
```
```   450   apply (induct k arbitrary: n)
```
```   451   apply (simp)
```
```   452   unfolding power_Suc mult_assoc
```
```   453   by (case_tac n, auto)
```
```   454
```
```   455 lemma X_power_mult_right_nth: "((f :: ('a::comm_ring_1) fps) * X^k) \$n = (if n < k then 0 else f \$ (n - k))"
```
```   456   by (metis X_power_mult_nth mult_commute)
```
```   457
```
```   458
```
```   459
```
```   460
```
```   461 subsection{* Formal Power series form a metric space *}
```
```   462
```
```   463 definition (in dist) ball_def: "ball x r = {y. dist y x < r}"
```
```   464 instantiation fps :: (comm_ring_1) dist
```
```   465 begin
```
```   466
```
```   467 definition dist_fps_def: "dist (a::'a fps) b = (if (\<exists>n. a\$n \<noteq> b\$n) then inverse (2 ^ The (leastP (\<lambda>n. a\$n \<noteq> b\$n))) else 0)"
```
```   468
```
```   469 lemma dist_fps_ge0: "dist (a::'a fps) b \<ge> 0"
```
```   470   by (simp add: dist_fps_def)
```
```   471
```
```   472 lemma dist_fps_sym: "dist (a::'a fps) b = dist b a"
```
```   473   apply (auto simp add: dist_fps_def)
```
```   474   apply (rule cong[OF refl, where x="(\<lambda>n\<Colon>nat. a \$ n \<noteq> b \$ n)"])
```
```   475   apply (rule ext)
```
```   476   by auto
```
```   477 instance ..
```
```   478 end
```
```   479
```
```   480 lemma fps_nonzero_least_unique: assumes a0: "a \<noteq> 0"
```
```   481   shows "\<exists>! n. leastP (\<lambda>n. a\$n \<noteq> 0) n"
```
```   482 proof-
```
```   483   from fps_nonzero_nth_minimal[of a] a0
```
```   484   obtain n where n: "a\$n \<noteq> 0" "\<forall>m < n. a\$m = 0" by blast
```
```   485   from n have ln: "leastP (\<lambda>n. a\$n \<noteq> 0) n"
```
```   486     by (auto simp add: leastP_def setge_def not_le[symmetric])
```
```   487   moreover
```
```   488   {fix m assume "leastP (\<lambda>n. a\$n \<noteq> 0) m"
```
```   489     then have "m = n" using ln
```
```   490       apply (auto simp add: leastP_def setge_def)
```
```   491       apply (erule allE[where x=n])
```
```   492       apply (erule allE[where x=m])
```
```   493       by simp}
```
```   494   ultimately show ?thesis by blast
```
```   495 qed
```
```   496
```
```   497 lemma fps_eq_least_unique: assumes ab: "(a::('a::ab_group_add) fps) \<noteq> b"
```
```   498   shows "\<exists>! n. leastP (\<lambda>n. a\$n \<noteq> b\$n) n"
```
```   499 using fps_nonzero_least_unique[of "a - b"] ab
```
```   500 by auto
```
```   501
```
```   502 instantiation fps :: (comm_ring_1) metric_space
```
```   503 begin
```
```   504
```
```   505 definition open_fps_def: "open (S :: 'a fps set) = (\<forall>a \<in> S. \<exists>r. r >0 \<and> ball a r \<subseteq> S)"
```
```   506
```
```   507 instance
```
```   508 proof
```
```   509   fix S :: "'a fps set"
```
```   510   show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   511     by (auto simp add: open_fps_def ball_def subset_eq)
```
```   512 next
```
```   513 {  fix a b :: "'a fps"
```
```   514   {assume ab: "a = b"
```
```   515    then have "\<not> (\<exists>n. a\$n \<noteq> b\$n)" by simp
```
```   516    then have "dist a b = 0" by (simp add: dist_fps_def)}
```
```   517  moreover
```
```   518  {assume d: "dist a b = 0"
```
```   519    then have "\<forall>n. a\$n = b\$n"
```
```   520      by - (rule ccontr, simp add: dist_fps_def)
```
```   521    then have "a = b" by (simp add: fps_eq_iff)}
```
```   522  ultimately show "dist a b =0 \<longleftrightarrow> a = b" by blast}
```
```   523 note th = this
```
```   524 from th have th'[simp]: "\<And>a::'a fps. dist a a = 0" by simp
```
```   525   fix a b c :: "'a fps"
```
```   526   {assume ab: "a = b" then have d0: "dist a b = 0"  unfolding th .
```
```   527     then have "dist a b \<le> dist a c + dist b c"
```
```   528       using dist_fps_ge0[of a c] dist_fps_ge0[of b c] by simp}
```
```   529   moreover
```
```   530   {assume c: "c = a \<or> c = b" then have "dist a b \<le> dist a c + dist b c"
```
```   531       by (cases "c=a", simp_all add: th dist_fps_sym) }
```
```   532   moreover
```
```   533   {assume ab: "a \<noteq> b" and ac: "a \<noteq> c" and bc: "b \<noteq> c"
```
```   534     let ?P = "\<lambda>a b n. a\$n \<noteq> b\$n"
```
```   535     from fps_eq_least_unique[OF ab] fps_eq_least_unique[OF ac]
```
```   536       fps_eq_least_unique[OF bc]
```
```   537     obtain nab nac nbc where nab: "leastP (?P a b) nab"
```
```   538       and nac: "leastP (?P a c) nac"
```
```   539       and nbc: "leastP (?P b c) nbc" by blast
```
```   540     from nab have nab': "\<And>m. m < nab \<Longrightarrow> a\$m = b\$m" "a\$nab \<noteq> b\$nab"
```
```   541       by (auto simp add: leastP_def setge_def)
```
```   542     from nac have nac': "\<And>m. m < nac \<Longrightarrow> a\$m = c\$m" "a\$nac \<noteq> c\$nac"
```
```   543       by (auto simp add: leastP_def setge_def)
```
```   544     from nbc have nbc': "\<And>m. m < nbc \<Longrightarrow> b\$m = c\$m" "b\$nbc \<noteq> c\$nbc"
```
```   545       by (auto simp add: leastP_def setge_def)
```
```   546
```
```   547     have th0: "\<And>(a::'a fps) b. a \<noteq> b \<longleftrightarrow> (\<exists>n. a\$n \<noteq> b\$n)"
```
```   548       by (simp add: fps_eq_iff)
```
```   549     from ab ac bc nab nac nbc
```
```   550     have dab: "dist a b = inverse (2 ^ nab)"
```
```   551       and dac: "dist a c = inverse (2 ^ nac)"
```
```   552       and dbc: "dist b c = inverse (2 ^ nbc)"
```
```   553       unfolding th0
```
```   554       apply (simp_all add: dist_fps_def)
```
```   555       apply (erule the1_equality[OF fps_eq_least_unique[OF ab]])
```
```   556       apply (erule the1_equality[OF fps_eq_least_unique[OF ac]])
```
```   557       by (erule the1_equality[OF fps_eq_least_unique[OF bc]])
```
```   558     from ab ac bc have nz: "dist a b \<noteq> 0" "dist a c \<noteq> 0" "dist b c \<noteq> 0"
```
```   559       unfolding th by simp_all
```
```   560     from nz have pos: "dist a b > 0" "dist a c > 0" "dist b c > 0"
```
```   561       using dist_fps_ge0[of a b] dist_fps_ge0[of a c] dist_fps_ge0[of b c]
```
```   562       by auto
```
```   563     have th1: "\<And>n. (2::real)^n >0" by auto
```
```   564     {assume h: "dist a b > dist a c + dist b c"
```
```   565       then have gt: "dist a b > dist a c" "dist a b > dist b c"
```
```   566         using pos by auto
```
```   567       from gt have gtn: "nab < nbc" "nab < nac"
```
```   568         unfolding dab dbc dac by (auto simp add: th1)
```
```   569       from nac'(1)[OF gtn(2)] nbc'(1)[OF gtn(1)]
```
```   570       have "a\$nab = b\$nab" by simp
```
```   571       with nab'(2) have False  by simp}
```
```   572     then have "dist a b \<le> dist a c + dist b c"
```
```   573       by (auto simp add: not_le[symmetric]) }
```
```   574   ultimately show "dist a b \<le> dist a c + dist b c" by blast
```
```   575 qed
```
```   576
```
```   577 end
```
```   578
```
```   579 text{* The infinite sums and justification of the notation in textbooks*}
```
```   580
```
```   581 lemma reals_power_lt_ex: assumes xp: "x > 0" and y1: "(y::real) > 1"
```
```   582   shows "\<exists>k>0. (1/y)^k < x"
```
```   583 proof-
```
```   584   have yp: "y > 0" using y1 by simp
```
```   585   from reals_Archimedean2[of "max 0 (- log y x) + 1"]
```
```   586   obtain k::nat where k: "real k > max 0 (- log y x) + 1" by blast
```
```   587   from k have kp: "k > 0" by simp
```
```   588   from k have "real k > - log y x" by simp
```
```   589   then have "ln y * real k > - ln x" unfolding log_def
```
```   590     using ln_gt_zero_iff[OF yp] y1
```
```   591     by (simp add: minus_divide_left field_simps del:minus_divide_left[symmetric])
```
```   592   then have "ln y * real k + ln x > 0" by simp
```
```   593   then have "exp (real k * ln y + ln x) > exp 0"
```
```   594     by (simp add: mult_ac)
```
```   595   then have "y ^ k * x > 1"
```
```   596     unfolding exp_zero exp_add exp_real_of_nat_mult
```
```   597     exp_ln[OF xp] exp_ln[OF yp] by simp
```
```   598   then have "x > (1/y)^k" using yp
```
```   599     by (simp add: field_simps nonzero_power_divide)
```
```   600   then show ?thesis using kp by blast
```
```   601 qed
```
```   602 lemma X_nth[simp]: "X\$n = (if n = 1 then 1 else 0)" by (simp add: X_def)
```
```   603 lemma X_power_nth[simp]: "(X^k) \$n = (if n = k then 1 else (0::'a::comm_ring_1))"
```
```   604   by (simp add: X_power_iff)
```
```   605
```
```   606
```
```   607 lemma fps_sum_rep_nth: "(setsum (%i. fps_const(a\$i)*X^i) {0..m})\$n = (if n \<le> m then a\$n else (0::'a::comm_ring_1))"
```
```   608   apply (auto simp add: fps_eq_iff fps_setsum_nth X_power_nth cond_application_beta cond_value_iff  cong del: if_weak_cong)
```
```   609   by (simp add: setsum_delta')
```
```   610
```
```   611 lemma fps_notation:
```
```   612   "(%n. setsum (%i. fps_const(a\$i) * X^i) {0..n}) ----> a" (is "?s ----> a")
```
```   613 proof-
```
```   614     {fix r:: real
```
```   615       assume rp: "r > 0"
```
```   616       have th0: "(2::real) > 1" by simp
```
```   617       from reals_power_lt_ex[OF rp th0]
```
```   618       obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0" by blast
```
```   619       {fix n::nat
```
```   620         assume nn0: "n \<ge> n0"
```
```   621         then have thnn0: "(1/2)^n <= (1/2 :: real)^n0"
```
```   622           by (auto intro: power_decreasing)
```
```   623         {assume "?s n = a" then have "dist (?s n) a < r"
```
```   624             unfolding dist_eq_0_iff[of "?s n" a, symmetric]
```
```   625             using rp by (simp del: dist_eq_0_iff)}
```
```   626         moreover
```
```   627         {assume neq: "?s n \<noteq> a"
```
```   628           from fps_eq_least_unique[OF neq]
```
```   629           obtain k where k: "leastP (\<lambda>i. ?s n \$ i \<noteq> a\$i) k" by blast
```
```   630           have th0: "\<And>(a::'a fps) b. a \<noteq> b \<longleftrightarrow> (\<exists>n. a\$n \<noteq> b\$n)"
```
```   631             by (simp add: fps_eq_iff)
```
```   632           from neq have dth: "dist (?s n) a = (1/2)^k"
```
```   633             unfolding th0 dist_fps_def
```
```   634             unfolding the1_equality[OF fps_eq_least_unique[OF neq], OF k]
```
```   635             by (auto simp add: inverse_eq_divide power_divide)
```
```   636
```
```   637           from k have kn: "k > n"
```
```   638             by (simp add: leastP_def setge_def fps_sum_rep_nth split:split_if_asm)
```
```   639           then have "dist (?s n) a < (1/2)^n" unfolding dth
```
```   640             by (auto intro: power_strict_decreasing)
```
```   641           also have "\<dots> <= (1/2)^n0" using nn0
```
```   642             by (auto intro: power_decreasing)
```
```   643           also have "\<dots> < r" using n0 by simp
```
```   644           finally have "dist (?s n) a < r" .}
```
```   645         ultimately have "dist (?s n) a < r" by blast}
```
```   646       then have "\<exists>n0. \<forall> n \<ge> n0. dist (?s n) a < r " by blast}
```
```   647     then show ?thesis  unfolding  LIMSEQ_def by blast
```
```   648   qed
```
```   649
```
```   650 subsection{* Inverses of formal power series *}
```
```   651
```
```   652 declare setsum_cong[fundef_cong]
```
```   653
```
```   654 instantiation fps :: ("{comm_monoid_add, inverse, times, uminus}") inverse
```
```   655 begin
```
```   656
```
```   657 fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" where
```
```   658   "natfun_inverse f 0 = inverse (f\$0)"
```
```   659 | "natfun_inverse f n = - inverse (f\$0) * setsum (\<lambda>i. f\$i * natfun_inverse f (n - i)) {1..n}"
```
```   660
```
```   661 definition fps_inverse_def:
```
```   662   "inverse f = (if f \$ 0 = 0 then 0 else Abs_fps (natfun_inverse f))"
```
```   663
```
```   664 definition fps_divide_def: "divide = (\<lambda>(f::'a fps) g. f * inverse g)"
```
```   665
```
```   666 instance ..
```
```   667
```
```   668 end
```
```   669
```
```   670 lemma fps_inverse_zero[simp]:
```
```   671   "inverse (0 :: 'a::{comm_monoid_add,inverse, times, uminus} fps) = 0"
```
```   672   by (simp add: fps_ext fps_inverse_def)
```
```   673
```
```   674 lemma fps_inverse_one[simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"
```
```   675   apply (auto simp add: expand_fps_eq fps_inverse_def)
```
```   676   by (case_tac n, auto)
```
```   677
```
```   678 lemma inverse_mult_eq_1 [intro]: assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```   679   shows "inverse f * f = 1"
```
```   680 proof-
```
```   681   have c: "inverse f * f = f * inverse f" by (simp add: mult_commute)
```
```   682   from f0 have ifn: "\<And>n. inverse f \$ n = natfun_inverse f n"
```
```   683     by (simp add: fps_inverse_def)
```
```   684   from f0 have th0: "(inverse f * f) \$ 0 = 1"
```
```   685     by (simp add: fps_mult_nth fps_inverse_def)
```
```   686   {fix n::nat assume np: "n >0 "
```
```   687     from np have eq: "{0..n} = {0} \<union> {1 .. n}" by auto
```
```   688     have d: "{0} \<inter> {1 .. n} = {}" by auto
```
```   689     have f: "finite {0::nat}" "finite {1..n}" by auto
```
```   690     from f0 np have th0: "- (inverse f\$n) =
```
```   691       (setsum (\<lambda>i. f\$i * natfun_inverse f (n - i)) {1..n}) / (f\$0)"
```
```   692       by (cases n, simp, simp add: divide_inverse fps_inverse_def)
```
```   693     from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]
```
```   694     have th1: "setsum (\<lambda>i. f\$i * natfun_inverse f (n - i)) {1..n} =
```
```   695       - (f\$0) * (inverse f)\$n"
```
```   696       by (simp add: field_simps)
```
```   697     have "(f * inverse f) \$ n = (\<Sum>i = 0..n. f \$i * natfun_inverse f (n - i))"
```
```   698       unfolding fps_mult_nth ifn ..
```
```   699     also have "\<dots> = f\$0 * natfun_inverse f n
```
```   700       + (\<Sum>i = 1..n. f\$i * natfun_inverse f (n-i))"
```
```   701       unfolding setsum_Un_disjoint[OF f d, unfolded eq[symmetric]]
```
```   702       by simp
```
```   703     also have "\<dots> = 0" unfolding th1 ifn by simp
```
```   704     finally have "(inverse f * f)\$n = 0" unfolding c . }
```
```   705   with th0 show ?thesis by (simp add: fps_eq_iff)
```
```   706 qed
```
```   707
```
```   708 lemma fps_inverse_0_iff[simp]: "(inverse f)\$0 = (0::'a::division_ring) \<longleftrightarrow> f\$0 = 0"
```
```   709   by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero)
```
```   710
```
```   711 lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f \$0 = 0"
```
```   712 proof-
```
```   713   {assume "f\$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)}
```
```   714   moreover
```
```   715   {assume h: "inverse f = 0" and c: "f \$0 \<noteq> 0"
```
```   716     from inverse_mult_eq_1[OF c] h have False by simp}
```
```   717   ultimately show ?thesis by blast
```
```   718 qed
```
```   719
```
```   720 lemma fps_inverse_idempotent[intro]: assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```   721   shows "inverse (inverse f) = f"
```
```   722 proof-
```
```   723   from f0 have if0: "inverse f \$ 0 \<noteq> 0" by simp
```
```   724   from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0]
```
```   725   have th0: "inverse f * f = inverse f * inverse (inverse f)"   by (simp add: mult_ac)
```
```   726   then show ?thesis using f0 unfolding mult_cancel_left by simp
```
```   727 qed
```
```   728
```
```   729 lemma fps_inverse_unique: assumes f0: "f\$0 \<noteq> (0::'a::field)" and fg: "f*g = 1"
```
```   730   shows "inverse f = g"
```
```   731 proof-
```
```   732   from inverse_mult_eq_1[OF f0] fg
```
```   733   have th0: "inverse f * f = g * f" by (simp add: mult_ac)
```
```   734   then show ?thesis using f0  unfolding mult_cancel_right
```
```   735     by (auto simp add: expand_fps_eq)
```
```   736 qed
```
```   737
```
```   738 lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
```
```   739   = Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
```
```   740   apply (rule fps_inverse_unique)
```
```   741   apply simp
```
```   742   apply (simp add: fps_eq_iff fps_mult_nth)
```
```   743 proof(clarsimp)
```
```   744   fix n::nat assume n: "n > 0"
```
```   745   let ?f = "\<lambda>i. if n = i then (1\<Colon>'a) else if n - i = 1 then - 1 else 0"
```
```   746   let ?g = "\<lambda>i. if i = n then 1 else if i=n - 1 then - 1 else 0"
```
```   747   let ?h = "\<lambda>i. if i=n - 1 then - 1 else 0"
```
```   748   have th1: "setsum ?f {0..n} = setsum ?g {0..n}"
```
```   749     by (rule setsum_cong2) auto
```
```   750   have th2: "setsum ?g {0..n - 1} = setsum ?h {0..n - 1}"
```
```   751     using n apply - by (rule setsum_cong2) auto
```
```   752   have eq: "{0 .. n} = {0.. n - 1} \<union> {n}" by auto
```
```   753   from n have d: "{0.. n - 1} \<inter> {n} = {}" by auto
```
```   754   have f: "finite {0.. n - 1}" "finite {n}" by auto
```
```   755   show "setsum ?f {0..n} = 0"
```
```   756     unfolding th1
```
```   757     apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
```
```   758     unfolding th2
```
```   759     by(simp add: setsum_delta)
```
```   760 qed
```
```   761
```
```   762 subsection{* Formal Derivatives, and the MacLaurin theorem around 0*}
```
```   763
```
```   764 definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f \$ (n + 1))"
```
```   765
```
```   766 lemma fps_deriv_nth[simp]: "fps_deriv f \$ n = of_nat (n +1) * f \$ (n+1)" by (simp add: fps_deriv_def)
```
```   767
```
```   768 lemma fps_deriv_linear[simp]: "fps_deriv (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = fps_const a * fps_deriv f + fps_const b * fps_deriv g"
```
```   769   unfolding fps_eq_iff fps_add_nth  fps_const_mult_left fps_deriv_nth by (simp add: field_simps)
```
```   770
```
```   771 lemma fps_deriv_mult[simp]:
```
```   772   fixes f :: "('a :: comm_ring_1) fps"
```
```   773   shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
```
```   774 proof-
```
```   775   let ?D = "fps_deriv"
```
```   776   {fix n::nat
```
```   777     let ?Zn = "{0 ..n}"
```
```   778     let ?Zn1 = "{0 .. n + 1}"
```
```   779     let ?f = "\<lambda>i. i + 1"
```
```   780     have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def)
```
```   781     have eq: "{1.. n+1} = ?f ` {0..n}" by auto
```
```   782     let ?g = "\<lambda>i. of_nat (i+1) * g \$ (i+1) * f \$ (n - i) +
```
```   783         of_nat (i+1)* f \$ (i+1) * g \$ (n - i)"
```
```   784     let ?h = "\<lambda>i. of_nat i * g \$ i * f \$ ((n+1) - i) +
```
```   785         of_nat i* f \$ i * g \$ ((n + 1) - i)"
```
```   786     {fix k assume k: "k \<in> {0..n}"
```
```   787       have "?h (k + 1) = ?g k" using k by auto}
```
```   788     note th0 = this
```
```   789     have eq': "{0..n +1}- {1 .. n+1} = {0}" by auto
```
```   790     have s0: "setsum (\<lambda>i. of_nat i * f \$ i * g \$ (n + 1 - i)) ?Zn1 = setsum (\<lambda>i. of_nat (n + 1 - i) * f \$ (n + 1 - i) * g \$ i) ?Zn1"
```
```   791       apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
```
```   792       apply (simp add: inj_on_def Ball_def)
```
```   793       apply presburger
```
```   794       apply (rule set_eqI)
```
```   795       apply (presburger add: image_iff)
```
```   796       by simp
```
```   797     have s1: "setsum (\<lambda>i. f \$ i * g \$ (n + 1 - i)) ?Zn1 = setsum (\<lambda>i. f \$ (n + 1 - i) * g \$ i) ?Zn1"
```
```   798       apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
```
```   799       apply (simp add: inj_on_def Ball_def)
```
```   800       apply presburger
```
```   801       apply (rule set_eqI)
```
```   802       apply (presburger add: image_iff)
```
```   803       by simp
```
```   804     have "(f * ?D g + ?D f * g)\$n = (?D g * f + ?D f * g)\$n" by (simp only: mult_commute)
```
```   805     also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
```
```   806       by (simp add: fps_mult_nth setsum_addf[symmetric])
```
```   807     also have "\<dots> = setsum ?h {1..n+1}"
```
```   808       using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto
```
```   809     also have "\<dots> = setsum ?h {0..n+1}"
```
```   810       apply (rule setsum_mono_zero_left)
```
```   811       apply simp
```
```   812       apply (simp add: subset_eq)
```
```   813       unfolding eq'
```
```   814       by simp
```
```   815     also have "\<dots> = (fps_deriv (f * g)) \$ n"
```
```   816       apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf)
```
```   817       unfolding s0 s1
```
```   818       unfolding setsum_addf[symmetric] setsum_right_distrib
```
```   819       apply (rule setsum_cong2)
```
```   820       by (auto simp add: of_nat_diff field_simps)
```
```   821     finally have "(f * ?D g + ?D f * g) \$ n = ?D (f*g) \$ n" .}
```
```   822   then show ?thesis unfolding fps_eq_iff by auto
```
```   823 qed
```
```   824
```
```   825 lemma fps_deriv_X[simp]: "fps_deriv X = 1"
```
```   826   by (simp add: fps_deriv_def X_def fps_eq_iff)
```
```   827
```
```   828 lemma fps_deriv_neg[simp]: "fps_deriv (- (f:: ('a:: comm_ring_1) fps)) = - (fps_deriv f)"
```
```   829   by (simp add: fps_eq_iff fps_deriv_def)
```
```   830 lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g"
```
```   831   using fps_deriv_linear[of 1 f 1 g] by simp
```
```   832
```
```   833 lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) - g) = fps_deriv f - fps_deriv g"
```
```   834   unfolding diff_minus by simp
```
```   835
```
```   836 lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"
```
```   837   by (simp add: fps_ext fps_deriv_def fps_const_def)
```
```   838
```
```   839 lemma fps_deriv_mult_const_left[simp]: "fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f"
```
```   840   by simp
```
```   841
```
```   842 lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"
```
```   843   by (simp add: fps_deriv_def fps_eq_iff)
```
```   844
```
```   845 lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
```
```   846   by (simp add: fps_deriv_def fps_eq_iff )
```
```   847
```
```   848 lemma fps_deriv_mult_const_right[simp]: "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c"
```
```   849   by simp
```
```   850
```
```   851 lemma fps_deriv_setsum: "fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: ('a::comm_ring_1) fps)) S"
```
```   852 proof-
```
```   853   {assume "\<not> finite S" hence ?thesis by simp}
```
```   854   moreover
```
```   855   {assume fS: "finite S"
```
```   856     have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
```
```   857   ultimately show ?thesis by blast
```
```   858 qed
```
```   859
```
```   860 lemma fps_deriv_eq_0_iff[simp]: "fps_deriv f = 0 \<longleftrightarrow> (f = fps_const (f\$0 :: 'a::{idom,semiring_char_0}))"
```
```   861 proof-
```
```   862   {assume "f= fps_const (f\$0)" hence "fps_deriv f = fps_deriv (fps_const (f\$0))" by simp
```
```   863     hence "fps_deriv f = 0" by simp }
```
```   864   moreover
```
```   865   {assume z: "fps_deriv f = 0"
```
```   866     hence "\<forall>n. (fps_deriv f)\$n = 0" by simp
```
```   867     hence "\<forall>n. f\$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)
```
```   868     hence "f = fps_const (f\$0)"
```
```   869       apply (clarsimp simp add: fps_eq_iff fps_const_def)
```
```   870       apply (erule_tac x="n - 1" in allE)
```
```   871       by simp}
```
```   872   ultimately show ?thesis by blast
```
```   873 qed
```
```   874
```
```   875 lemma fps_deriv_eq_iff:
```
```   876   fixes f:: "('a::{idom,semiring_char_0}) fps"
```
```   877   shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f\$0 - g\$0) + g)"
```
```   878 proof-
```
```   879   have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0" by simp
```
```   880   also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f-g)\$0)" unfolding fps_deriv_eq_0_iff ..
```
```   881   finally show ?thesis by (simp add: field_simps)
```
```   882 qed
```
```   883
```
```   884 lemma fps_deriv_eq_iff_ex: "(fps_deriv f = fps_deriv g) \<longleftrightarrow> (\<exists>(c::'a::{idom,semiring_char_0}). f = fps_const c + g)"
```
```   885   apply auto unfolding fps_deriv_eq_iff by blast
```
```   886
```
```   887
```
```   888 fun fps_nth_deriv :: "nat \<Rightarrow> ('a::semiring_1) fps \<Rightarrow> 'a fps" where
```
```   889   "fps_nth_deriv 0 f = f"
```
```   890 | "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"
```
```   891
```
```   892 lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"
```
```   893   by (induct n arbitrary: f, auto)
```
```   894
```
```   895 lemma fps_nth_deriv_linear[simp]: "fps_nth_deriv n (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g"
```
```   896   by (induct n arbitrary: f g, auto simp add: fps_nth_deriv_commute)
```
```   897
```
```   898 lemma fps_nth_deriv_neg[simp]: "fps_nth_deriv n (- (f:: ('a:: comm_ring_1) fps)) = - (fps_nth_deriv n f)"
```
```   899   by (induct n arbitrary: f, simp_all)
```
```   900
```
```   901 lemma fps_nth_deriv_add[simp]: "fps_nth_deriv n ((f:: ('a::comm_ring_1) fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g"
```
```   902   using fps_nth_deriv_linear[of n 1 f 1 g] by simp
```
```   903
```
```   904 lemma fps_nth_deriv_sub[simp]: "fps_nth_deriv n ((f:: ('a::comm_ring_1) fps) - g) = fps_nth_deriv n f - fps_nth_deriv n g"
```
```   905   unfolding diff_minus fps_nth_deriv_add by simp
```
```   906
```
```   907 lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"
```
```   908   by (induct n, simp_all )
```
```   909
```
```   910 lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"
```
```   911   by (induct n, simp_all )
```
```   912
```
```   913 lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"
```
```   914   by (cases n, simp_all)
```
```   915
```
```   916 lemma fps_nth_deriv_mult_const_left[simp]: "fps_nth_deriv n (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_nth_deriv n f"
```
```   917   using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp
```
```   918
```
```   919 lemma fps_nth_deriv_mult_const_right[simp]: "fps_nth_deriv n (f * fps_const (c::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c"
```
```   920   using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute)
```
```   921
```
```   922 lemma fps_nth_deriv_setsum: "fps_nth_deriv n (setsum f S) = setsum (\<lambda>i. fps_nth_deriv n (f i :: ('a::comm_ring_1) fps)) S"
```
```   923 proof-
```
```   924   {assume "\<not> finite S" hence ?thesis by simp}
```
```   925   moreover
```
```   926   {assume fS: "finite S"
```
```   927     have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
```
```   928   ultimately show ?thesis by blast
```
```   929 qed
```
```   930
```
```   931 lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) \$ 0 = of_nat (fact k) * f\$(k)"
```
```   932   by (induct k arbitrary: f) (auto simp add: field_simps of_nat_mult)
```
```   933
```
```   934 subsection {* Powers*}
```
```   935
```
```   936 lemma fps_power_zeroth_eq_one: "a\$0 =1 \<Longrightarrow> a^n \$ 0 = (1::'a::semiring_1)"
```
```   937   by (induct n, auto simp add: expand_fps_eq fps_mult_nth)
```
```   938
```
```   939 lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)\$0 =1 \<Longrightarrow> a^n \$ 1 = of_nat n * a\$1"
```
```   940 proof(induct n)
```
```   941   case 0 thus ?case by simp
```
```   942 next
```
```   943   case (Suc n)
```
```   944   note h = Suc.hyps[OF `a\$0 = 1`]
```
```   945   show ?case unfolding power_Suc fps_mult_nth
```
```   946     using h `a\$0 = 1`  fps_power_zeroth_eq_one[OF `a\$0=1`] by (simp add: field_simps)
```
```   947 qed
```
```   948
```
```   949 lemma startsby_one_power:"a \$ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n \$ 0 = 1"
```
```   950   by (induct n, auto simp add: fps_mult_nth)
```
```   951
```
```   952 lemma startsby_zero_power:"a \$0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n \$0 = 0"
```
```   953   by (induct n, auto simp add: fps_mult_nth)
```
```   954
```
```   955 lemma startsby_power:"a \$0 = (v::'a::{comm_ring_1}) \<Longrightarrow> a^n \$0 = v^n"
```
```   956   by (induct n, auto simp add: fps_mult_nth power_Suc)
```
```   957
```
```   958 lemma startsby_zero_power_iff[simp]:
```
```   959   "a^n \$0 = (0::'a::{idom}) \<longleftrightarrow> (n \<noteq> 0 \<and> a\$0 = 0)"
```
```   960 apply (rule iffI)
```
```   961 apply (induct n, auto simp add: power_Suc fps_mult_nth)
```
```   962 by (rule startsby_zero_power, simp_all)
```
```   963
```
```   964 lemma startsby_zero_power_prefix:
```
```   965   assumes a0: "a \$0 = (0::'a::idom)"
```
```   966   shows "\<forall>n < k. a ^ k \$ n = 0"
```
```   967   using a0
```
```   968 proof(induct k rule: nat_less_induct)
```
```   969   fix k assume H: "\<forall>m<k. a \$0 =  0 \<longrightarrow> (\<forall>n<m. a ^ m \$ n = 0)" and a0: "a \$0 = (0\<Colon>'a)"
```
```   970   let ?ths = "\<forall>m<k. a ^ k \$ m = 0"
```
```   971   {assume "k = 0" then have ?ths by simp}
```
```   972   moreover
```
```   973   {fix l assume k: "k = Suc l"
```
```   974     {fix m assume mk: "m < k"
```
```   975       {assume "m=0" hence "a^k \$ m = 0" using startsby_zero_power[of a k] k a0
```
```   976           by simp}
```
```   977       moreover
```
```   978       {assume m0: "m \<noteq> 0"
```
```   979         have "a ^k \$ m = (a^l * a) \$m" by (simp add: k power_Suc mult_commute)
```
```   980         also have "\<dots> = (\<Sum>i = 0..m. a ^ l \$ i * a \$ (m - i))" by (simp add: fps_mult_nth)
```
```   981         also have "\<dots> = 0" apply (rule setsum_0')
```
```   982           apply auto
```
```   983           apply (case_tac "aa = m")
```
```   984           using a0
```
```   985           apply simp
```
```   986           apply (rule H[rule_format])
```
```   987           using a0 k mk by auto
```
```   988         finally have "a^k \$ m = 0" .}
```
```   989     ultimately have "a^k \$ m = 0" by blast}
```
```   990     hence ?ths by blast}
```
```   991   ultimately show ?ths by (cases k, auto)
```
```   992 qed
```
```   993
```
```   994 lemma startsby_zero_setsum_depends:
```
```   995   assumes a0: "a \$0 = (0::'a::idom)" and kn: "n \<ge> k"
```
```   996   shows "setsum (\<lambda>i. (a ^ i)\$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)\$k) {0 .. k}"
```
```   997   apply (rule setsum_mono_zero_right)
```
```   998   using kn apply auto
```
```   999   apply (rule startsby_zero_power_prefix[rule_format, OF a0])
```
```  1000   by arith
```
```  1001
```
```  1002 lemma startsby_zero_power_nth_same: assumes a0: "a\$0 = (0::'a::{idom})"
```
```  1003   shows "a^n \$ n = (a\$1) ^ n"
```
```  1004 proof(induct n)
```
```  1005   case 0 thus ?case by (simp add: power_0)
```
```  1006 next
```
```  1007   case (Suc n)
```
```  1008   have "a ^ Suc n \$ (Suc n) = (a^n * a)\$(Suc n)" by (simp add: field_simps power_Suc)
```
```  1009   also have "\<dots> = setsum (\<lambda>i. a^n\$i * a \$ (Suc n - i)) {0.. Suc n}" by (simp add: fps_mult_nth)
```
```  1010   also have "\<dots> = setsum (\<lambda>i. a^n\$i * a \$ (Suc n - i)) {n .. Suc n}"
```
```  1011     apply (rule setsum_mono_zero_right)
```
```  1012     apply simp
```
```  1013     apply clarsimp
```
```  1014     apply clarsimp
```
```  1015     apply (rule startsby_zero_power_prefix[rule_format, OF a0])
```
```  1016     apply arith
```
```  1017     done
```
```  1018   also have "\<dots> = a^n \$ n * a\$1" using a0 by simp
```
```  1019   finally show ?case using Suc.hyps by (simp add: power_Suc)
```
```  1020 qed
```
```  1021
```
```  1022 lemma fps_inverse_power:
```
```  1023   fixes a :: "('a::{field}) fps"
```
```  1024   shows "inverse (a^n) = inverse a ^ n"
```
```  1025 proof-
```
```  1026   {assume a0: "a\$0 = 0"
```
```  1027     hence eq: "inverse a = 0" by (simp add: fps_inverse_def)
```
```  1028     {assume "n = 0" hence ?thesis by simp}
```
```  1029     moreover
```
```  1030     {assume n: "n > 0"
```
```  1031       from startsby_zero_power[OF a0 n] eq a0 n have ?thesis
```
```  1032         by (simp add: fps_inverse_def)}
```
```  1033     ultimately have ?thesis by blast}
```
```  1034   moreover
```
```  1035   {assume a0: "a\$0 \<noteq> 0"
```
```  1036     have ?thesis
```
```  1037       apply (rule fps_inverse_unique)
```
```  1038       apply (simp add: a0)
```
```  1039       unfolding power_mult_distrib[symmetric]
```
```  1040       apply (rule ssubst[where t = "a * inverse a" and s= 1])
```
```  1041       apply simp_all
```
```  1042       apply (subst mult_commute)
```
```  1043       by (rule inverse_mult_eq_1[OF a0])}
```
```  1044   ultimately show ?thesis by blast
```
```  1045 qed
```
```  1046
```
```  1047 lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n - 1)"
```
```  1048   apply (induct n, auto simp add: power_Suc field_simps fps_const_add[symmetric] simp del: fps_const_add)
```
```  1049   by (case_tac n, auto simp add: power_Suc field_simps)
```
```  1050
```
```  1051 lemma fps_inverse_deriv:
```
```  1052   fixes a:: "('a :: field) fps"
```
```  1053   assumes a0: "a\$0 \<noteq> 0"
```
```  1054   shows "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2"
```
```  1055 proof-
```
```  1056   from inverse_mult_eq_1[OF a0]
```
```  1057   have "fps_deriv (inverse a * a) = 0" by simp
```
```  1058   hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp
```
```  1059   hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0"  by simp
```
```  1060   with inverse_mult_eq_1[OF a0]
```
```  1061   have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0"
```
```  1062     unfolding power2_eq_square
```
```  1063     apply (simp add: field_simps)
```
```  1064     by (simp add: mult_assoc[symmetric])
```
```  1065   hence "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * inverse a ^ 2 = 0 - fps_deriv a * inverse a ^ 2"
```
```  1066     by simp
```
```  1067   then show "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2" by (simp add: field_simps)
```
```  1068 qed
```
```  1069
```
```  1070 lemma fps_inverse_mult:
```
```  1071   fixes a::"('a :: field) fps"
```
```  1072   shows "inverse (a * b) = inverse a * inverse b"
```
```  1073 proof-
```
```  1074   {assume a0: "a\$0 = 0" hence ab0: "(a*b)\$0 = 0" by (simp add: fps_mult_nth)
```
```  1075     from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all
```
```  1076     have ?thesis unfolding th by simp}
```
```  1077   moreover
```
```  1078   {assume b0: "b\$0 = 0" hence ab0: "(a*b)\$0 = 0" by (simp add: fps_mult_nth)
```
```  1079     from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all
```
```  1080     have ?thesis unfolding th by simp}
```
```  1081   moreover
```
```  1082   {assume a0: "a\$0 \<noteq> 0" and b0: "b\$0 \<noteq> 0"
```
```  1083     from a0 b0 have ab0:"(a*b) \$ 0 \<noteq> 0" by (simp  add: fps_mult_nth)
```
```  1084     from inverse_mult_eq_1[OF ab0]
```
```  1085     have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp
```
```  1086     then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b"
```
```  1087       by (simp add: field_simps)
```
```  1088     then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp}
```
```  1089 ultimately show ?thesis by blast
```
```  1090 qed
```
```  1091
```
```  1092 lemma fps_inverse_deriv':
```
```  1093   fixes a:: "('a :: field) fps"
```
```  1094   assumes a0: "a\$0 \<noteq> 0"
```
```  1095   shows "fps_deriv (inverse a) = - fps_deriv a / a ^ 2"
```
```  1096   using fps_inverse_deriv[OF a0]
```
```  1097   unfolding power2_eq_square fps_divide_def
```
```  1098     fps_inverse_mult by simp
```
```  1099
```
```  1100 lemma inverse_mult_eq_1': assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```  1101   shows "f * inverse f= 1"
```
```  1102   by (metis mult_commute inverse_mult_eq_1 f0)
```
```  1103
```
```  1104 lemma fps_divide_deriv:   fixes a:: "('a :: field) fps"
```
```  1105   assumes a0: "b\$0 \<noteq> 0"
```
```  1106   shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b ^ 2"
```
```  1107   using fps_inverse_deriv[OF a0]
```
```  1108   by (simp add: fps_divide_def field_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])
```
```  1109
```
```  1110
```
```  1111 lemma fps_inverse_gp': "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
```
```  1112   = 1 - X"
```
```  1113   by (simp add: fps_inverse_gp fps_eq_iff X_def)
```
```  1114
```
```  1115 lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)"
```
```  1116   by (cases "n", simp_all)
```
```  1117
```
```  1118
```
```  1119 lemma fps_inverse_X_plus1:
```
```  1120   "inverse (1 + X) = Abs_fps (\<lambda>n. (- (1::'a::{field})) ^ n)" (is "_ = ?r")
```
```  1121 proof-
```
```  1122   have eq: "(1 + X) * ?r = 1"
```
```  1123     unfolding minus_one_power_iff
```
```  1124     by (auto simp add: field_simps fps_eq_iff)
```
```  1125   show ?thesis by (auto simp add: eq intro: fps_inverse_unique)
```
```  1126 qed
```
```  1127
```
```  1128
```
```  1129 subsection{* Integration *}
```
```  1130
```
```  1131 definition
```
```  1132   fps_integral :: "'a::field_char_0 fps \<Rightarrow> 'a \<Rightarrow> 'a fps" where
```
```  1133   "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a\$(n - 1) / of_nat n))"
```
```  1134
```
```  1135 lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a"
```
```  1136   unfolding fps_integral_def fps_deriv_def
```
```  1137   by (simp add: fps_eq_iff del: of_nat_Suc)
```
```  1138
```
```  1139 lemma fps_integral_linear:
```
```  1140   "fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) =
```
```  1141     fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0"
```
```  1142   (is "?l = ?r")
```
```  1143 proof-
```
```  1144   have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral)
```
```  1145   moreover have "?l\$0 = ?r\$0" by (simp add: fps_integral_def)
```
```  1146   ultimately show ?thesis
```
```  1147     unfolding fps_deriv_eq_iff by auto
```
```  1148 qed
```
```  1149
```
```  1150 subsection {* Composition of FPSs *}
```
```  1151 definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where
```
```  1152   fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a\$i * (b^i\$n)) {0..n})"
```
```  1153
```
```  1154 lemma fps_compose_nth: "(a oo b)\$n = setsum (\<lambda>i. a\$i * (b^i\$n)) {0..n}" by (simp add: fps_compose_def)
```
```  1155
```
```  1156 lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)"
```
```  1157   by (simp add: fps_ext fps_compose_def mult_delta_right setsum_delta')
```
```  1158
```
```  1159 lemma fps_const_compose[simp]:
```
```  1160   "fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)"
```
```  1161   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta)
```
```  1162
```
```  1163 lemma number_of_compose[simp]: "(number_of k::('a::{comm_ring_1}) fps) oo b = number_of k"
```
```  1164   unfolding number_of_fps_const by simp
```
```  1165
```
```  1166 lemma X_fps_compose_startby0[simp]: "a\$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)"
```
```  1167   by (simp add: fps_eq_iff fps_compose_def mult_delta_left setsum_delta
```
```  1168                 power_Suc not_le)
```
```  1169
```
```  1170
```
```  1171 subsection {* Rules from Herbert Wilf's Generatingfunctionology*}
```
```  1172
```
```  1173 subsubsection {* Rule 1 *}
```
```  1174   (* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
```
```  1175
```
```  1176 lemma fps_power_mult_eq_shift:
```
```  1177   "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) = Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: comm_ring_1) * X^i) {0 .. k}" (is "?lhs = ?rhs")
```
```  1178 proof-
```
```  1179   {fix n:: nat
```
```  1180     have "?lhs \$ n = (if n < Suc k then 0 else a n)"
```
```  1181       unfolding X_power_mult_nth by auto
```
```  1182     also have "\<dots> = ?rhs \$ n"
```
```  1183     proof(induct k)
```
```  1184       case 0 thus ?case by (simp add: fps_setsum_nth power_Suc)
```
```  1185     next
```
```  1186       case (Suc k)
```
```  1187       note th = Suc.hyps[symmetric]
```
```  1188       have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. Suc k})\$n = (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) \$ n" by (simp add: field_simps)
```
```  1189       also  have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)\$n"
```
```  1190         using th
```
```  1191         unfolding fps_sub_nth by simp
```
```  1192       also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)"
```
```  1193         unfolding X_power_mult_right_nth
```
```  1194         apply (auto simp add: not_less fps_const_def)
```
```  1195         apply (rule cong[of a a, OF refl])
```
```  1196         by arith
```
```  1197       finally show ?case by simp
```
```  1198     qed
```
```  1199     finally have "?lhs \$ n = ?rhs \$ n"  .}
```
```  1200   then show ?thesis by (simp add: fps_eq_iff)
```
```  1201 qed
```
```  1202
```
```  1203 subsubsection{* Rule 2*}
```
```  1204
```
```  1205   (* We can not reach the form of Wilf, but still near to it using rewrite rules*)
```
```  1206   (* If f reprents {a_n} and P is a polynomial, then
```
```  1207         P(xD) f represents {P(n) a_n}*)
```
```  1208
```
```  1209 definition "XD = op * X o fps_deriv"
```
```  1210
```
```  1211 lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)"
```
```  1212   by (simp add: XD_def field_simps)
```
```  1213
```
```  1214 lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a"
```
```  1215   by (simp add: XD_def field_simps)
```
```  1216
```
```  1217 lemma XD_linear[simp]: "XD (fps_const c * a + fps_const d * b) = fps_const c * XD a + fps_const d * XD (b :: ('a::comm_ring_1) fps)"
```
```  1218   by simp
```
```  1219
```
```  1220 lemma XDN_linear:
```
```  1221   "(XD ^^ n) (fps_const c * a + fps_const d * b) = fps_const c * (XD ^^ n) a + fps_const d * (XD ^^ n) (b :: ('a::comm_ring_1) fps)"
```
```  1222   by (induct n, simp_all)
```
```  1223
```
```  1224 lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a\$n)" by (simp add: fps_eq_iff)
```
```  1225
```
```  1226
```
```  1227 lemma fps_mult_XD_shift:
```
```  1228   "(XD ^^ k) (a:: ('a::{comm_ring_1}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a\$n)"
```
```  1229   by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff field_simps del: One_nat_def)
```
```  1230
```
```  1231 subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}
```
```  1232 subsubsection{* Rule 5 --- summation and "division" by (1 - X)*}
```
```  1233
```
```  1234 lemma fps_divide_X_minus1_setsum_lemma:
```
```  1235   "a = ((1::('a::comm_ring_1) fps) - X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a \$ i) {0..n})"
```
```  1236 proof-
```
```  1237   let ?X = "X::('a::comm_ring_1) fps"
```
```  1238   let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a \$ i) {0..n})"
```
```  1239   have th0: "\<And>i. (1 - (X::'a fps)) \$ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)" by simp
```
```  1240   {fix n:: nat
```
```  1241     {assume "n=0" hence "a\$n = ((1 - ?X) * ?sa) \$ n"
```
```  1242         by (simp add: fps_mult_nth)}
```
```  1243     moreover
```
```  1244     {assume n0: "n \<noteq> 0"
```
```  1245       then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1}\<union>{2..n} = {1..n}"
```
```  1246         "{0..n - 1}\<union>{n} = {0..n}"
```
```  1247         by (auto simp: set_eq_iff)
```
```  1248       have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}"
```
```  1249         "{0..n - 1}\<inter>{n} ={}" using n0 by simp_all
```
```  1250       have f: "finite {0}" "finite {1}" "finite {2 .. n}"
```
```  1251         "finite {0 .. n - 1}" "finite {n}" by simp_all
```
```  1252     have "((1 - ?X) * ?sa) \$ n = setsum (\<lambda>i. (1 - ?X)\$ i * ?sa \$ (n - i)) {0 .. n}"
```
```  1253       by (simp add: fps_mult_nth)
```
```  1254     also have "\<dots> = a\$n" unfolding th0
```
```  1255       unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
```
```  1256       unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)]
```
```  1257       apply (simp)
```
```  1258       unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)]
```
```  1259       by simp
```
```  1260     finally have "a\$n = ((1 - ?X) * ?sa) \$ n" by simp}
```
```  1261   ultimately have "a\$n = ((1 - ?X) * ?sa) \$ n" by blast}
```
```  1262 then show ?thesis
```
```  1263   unfolding fps_eq_iff by blast
```
```  1264 qed
```
```  1265
```
```  1266 lemma fps_divide_X_minus1_setsum:
```
```  1267   "a /((1::('a::field) fps) - X)  = Abs_fps (\<lambda>n. setsum (\<lambda>i. a \$ i) {0..n})"
```
```  1268 proof-
```
```  1269   let ?X = "1 - (X::('a::field) fps)"
```
```  1270   have th0: "?X \$ 0 \<noteq> 0" by simp
```
```  1271   have "a /?X = ?X *  Abs_fps (\<lambda>n\<Colon>nat. setsum (op \$ a) {0..n}) * inverse ?X"
```
```  1272     using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0
```
```  1273     by (simp add: fps_divide_def mult_assoc)
```
```  1274   also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n\<Colon>nat. setsum (op \$ a) {0..n}) "
```
```  1275     by (simp add: mult_ac)
```
```  1276   finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0])
```
```  1277 qed
```
```  1278
```
```  1279 subsubsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary
```
```  1280   finite product of FPS, also the relvant instance of powers of a FPS*}
```
```  1281
```
```  1282 definition "natpermute n k = {l:: nat list. length l = k \<and> foldl op + 0 l = n}"
```
```  1283
```
```  1284 lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
```
```  1285   apply (auto simp add: natpermute_def)
```
```  1286   apply (case_tac x, auto)
```
```  1287   done
```
```  1288
```
```  1289 lemma foldl_add_start0:
```
```  1290   "foldl op + x xs = x + foldl op + (0::nat) xs"
```
```  1291   apply (induct xs arbitrary: x)
```
```  1292   apply simp
```
```  1293   unfolding foldl.simps
```
```  1294   apply atomize
```
```  1295   apply (subgoal_tac "\<forall>x\<Colon>nat. foldl op + x xs = x + foldl op + (0\<Colon>nat) xs")
```
```  1296   apply (erule_tac x="x + a" in allE)
```
```  1297   apply (erule_tac x="a" in allE)
```
```  1298   apply simp
```
```  1299   apply assumption
```
```  1300   done
```
```  1301
```
```  1302 lemma foldl_add_append: "foldl op + (x::nat) (xs@ys) = foldl op + x xs + foldl op + 0 ys"
```
```  1303   apply (induct ys arbitrary: x xs)
```
```  1304   apply auto
```
```  1305   apply (subst (2) foldl_add_start0)
```
```  1306   apply simp
```
```  1307   apply (subst (2) foldl_add_start0)
```
```  1308   by simp
```
```  1309
```
```  1310 lemma foldl_add_setsum: "foldl op + (x::nat) xs = x + setsum (nth xs) {0..<length xs}"
```
```  1311 proof(induct xs arbitrary: x)
```
```  1312   case Nil thus ?case by simp
```
```  1313 next
```
```  1314   case (Cons a as x)
```
```  1315   have eq: "setsum (op ! (a#as)) {1..<length (a#as)} = setsum (op ! as) {0..<length as}"
```
```  1316     apply (rule setsum_reindex_cong [where f=Suc])
```
```  1317     by (simp_all add: inj_on_def)
```
```  1318   have f: "finite {0}" "finite {1 ..< length (a#as)}" by simp_all
```
```  1319   have d: "{0} \<inter> {1..<length (a#as)} = {}" by simp
```
```  1320   have seq: "{0} \<union> {1..<length(a#as)} = {0 ..<length (a#as)}" by auto
```
```  1321   have "foldl op + x (a#as) = x + foldl op + a as "
```
```  1322     apply (subst foldl_add_start0)    by simp
```
```  1323   also have "\<dots> = x + a + setsum (op ! as) {0..<length as}" unfolding Cons.hyps by simp
```
```  1324   also have "\<dots> = x + setsum (op ! (a#as)) {0..<length (a#as)}"
```
```  1325     unfolding eq[symmetric]
```
```  1326     unfolding setsum_Un_disjoint[OF f d, unfolded seq]
```
```  1327     by simp
```
```  1328   finally show ?case  .
```
```  1329 qed
```
```  1330
```
```  1331
```
```  1332 lemma append_natpermute_less_eq:
```
```  1333   assumes h: "xs@ys \<in> natpermute n k" shows "foldl op + 0 xs \<le> n" and "foldl op + 0 ys \<le> n"
```
```  1334 proof-
```
```  1335   {from h have "foldl op + 0 (xs@ ys) = n" by (simp add: natpermute_def)
```
```  1336     hence "foldl op + 0 xs + foldl op + 0 ys = n" unfolding foldl_add_append .}
```
```  1337   note th = this
```
```  1338   {from th show "foldl op + 0 xs \<le> n" by simp}
```
```  1339   {from th show "foldl op + 0 ys \<le> n" by simp}
```
```  1340 qed
```
```  1341
```
```  1342 lemma natpermute_split:
```
```  1343   assumes mn: "h \<le> k"
```
```  1344   shows "natpermute n k = (\<Union>m \<in>{0..n}. {l1 @ l2 |l1 l2. l1 \<in> natpermute m h \<and> l2 \<in> natpermute (n - m) (k - h)})" (is "?L = ?R" is "?L = (\<Union>m \<in>{0..n}. ?S m)")
```
```  1345 proof-
```
```  1346   {fix l assume l: "l \<in> ?R"
```
```  1347     from l obtain m xs ys where h: "m \<in> {0..n}" and xs: "xs \<in> natpermute m h" and ys: "ys \<in> natpermute (n - m) (k - h)"  and leq: "l = xs@ys" by blast
```
```  1348     from xs have xs': "foldl op + 0 xs = m" by (simp add: natpermute_def)
```
```  1349     from ys have ys': "foldl op + 0 ys = n - m" by (simp add: natpermute_def)
```
```  1350     have "l \<in> ?L" using leq xs ys h
```
```  1351       apply simp
```
```  1352       apply (clarsimp simp add: natpermute_def simp del: foldl_append)
```
```  1353       apply (simp add: foldl_add_append[unfolded foldl_append])
```
```  1354       unfolding xs' ys'
```
```  1355       using mn xs ys
```
```  1356       unfolding natpermute_def by simp}
```
```  1357   moreover
```
```  1358   {fix l assume l: "l \<in> natpermute n k"
```
```  1359     let ?xs = "take h l"
```
```  1360     let ?ys = "drop h l"
```
```  1361     let ?m = "foldl op + 0 ?xs"
```
```  1362     from l have ls: "foldl op + 0 (?xs @ ?ys) = n" by (simp add: natpermute_def)
```
```  1363     have xs: "?xs \<in> natpermute ?m h" using l mn by (simp add: natpermute_def)
```
```  1364     have ys: "?ys \<in> natpermute (n - ?m) (k - h)" using l mn ls[unfolded foldl_add_append]
```
```  1365       by (simp add: natpermute_def)
```
```  1366     from ls have m: "?m \<in> {0..n}"  unfolding foldl_add_append by simp
```
```  1367     from xs ys ls have "l \<in> ?R"
```
```  1368       apply auto
```
```  1369       apply (rule bexI[where x = "?m"])
```
```  1370       apply (rule exI[where x = "?xs"])
```
```  1371       apply (rule exI[where x = "?ys"])
```
```  1372       using ls l unfolding foldl_add_append
```
```  1373       by (auto simp add: natpermute_def)}
```
```  1374   ultimately show ?thesis by blast
```
```  1375 qed
```
```  1376
```
```  1377 lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"
```
```  1378   by (auto simp add: natpermute_def)
```
```  1379 lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"
```
```  1380   apply (auto simp add: set_replicate_conv_if natpermute_def)
```
```  1381   apply (rule nth_equalityI)
```
```  1382   by simp_all
```
```  1383
```
```  1384 lemma natpermute_finite: "finite (natpermute n k)"
```
```  1385 proof(induct k arbitrary: n)
```
```  1386   case 0 thus ?case
```
```  1387     apply (subst natpermute_split[of 0 0, simplified])
```
```  1388     by (simp add: natpermute_0)
```
```  1389 next
```
```  1390   case (Suc k)
```
```  1391   then show ?case unfolding natpermute_split[of k "Suc k", simplified]
```
```  1392     apply -
```
```  1393     apply (rule finite_UN_I)
```
```  1394     apply simp
```
```  1395     unfolding One_nat_def[symmetric] natlist_trivial_1
```
```  1396     apply simp
```
```  1397     done
```
```  1398 qed
```
```  1399
```
```  1400 lemma natpermute_contain_maximal:
```
```  1401   "{xs \<in> natpermute n (k+1). n \<in> set xs} = UNION {0 .. k} (\<lambda>i. {(replicate (k+1) 0) [i:=n]})"
```
```  1402   (is "?A = ?B")
```
```  1403 proof-
```
```  1404   {fix xs assume H: "xs \<in> natpermute n (k+1)" and n: "n \<in> set xs"
```
```  1405     from n obtain i where i: "i \<in> {0.. k}" "xs!i = n" using H
```
```  1406       unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def)
```
```  1407     have eqs: "({0..k} - {i}) \<union> {i} = {0..k}" using i by auto
```
```  1408     have f: "finite({0..k} - {i})" "finite {i}" by auto
```
```  1409     have d: "({0..k} - {i}) \<inter> {i} = {}" using i by auto
```
```  1410     from H have "n = setsum (nth xs) {0..k}" apply (simp add: natpermute_def)
```
```  1411       unfolding foldl_add_setsum by (auto simp add: atLeastLessThanSuc_atLeastAtMost)
```
```  1412     also have "\<dots> = n + setsum (nth xs) ({0..k} - {i})"
```
```  1413       unfolding setsum_Un_disjoint[OF f d, unfolded eqs] using i by simp
```
```  1414     finally have zxs: "\<forall> j\<in> {0..k} - {i}. xs!j = 0" by auto
```
```  1415     from H have xsl: "length xs = k+1" by (simp add: natpermute_def)
```
```  1416     from i have i': "i < length (replicate (k+1) 0)"   "i < k+1"
```
```  1417       unfolding length_replicate  by arith+
```
```  1418     have "xs = replicate (k+1) 0 [i := n]"
```
```  1419       apply (rule nth_equalityI)
```
```  1420       unfolding xsl length_list_update length_replicate
```
```  1421       apply simp
```
```  1422       apply clarify
```
```  1423       unfolding nth_list_update[OF i'(1)]
```
```  1424       using i zxs
```
```  1425       by (case_tac "ia=i", auto simp del: replicate.simps)
```
```  1426     then have "xs \<in> ?B" using i by blast}
```
```  1427   moreover
```
```  1428   {fix i assume i: "i \<in> {0..k}"
```
```  1429     let ?xs = "replicate (k+1) 0 [i:=n]"
```
```  1430     have nxs: "n \<in> set ?xs"
```
```  1431       apply (rule set_update_memI) using i by simp
```
```  1432     have xsl: "length ?xs = k+1" by (simp only: length_replicate length_list_update)
```
```  1433     have "foldl op + 0 ?xs = setsum (nth ?xs) {0..<k+1}"
```
```  1434       unfolding foldl_add_setsum add_0 length_replicate length_list_update ..
```
```  1435     also have "\<dots> = setsum (\<lambda>j. if j = i then n else 0) {0..< k+1}"
```
```  1436       apply (rule setsum_cong2) by (simp del: replicate.simps)
```
```  1437     also have "\<dots> = n" using i by (simp add: setsum_delta)
```
```  1438     finally
```
```  1439     have "?xs \<in> natpermute n (k+1)" using xsl unfolding natpermute_def mem_Collect_eq
```
```  1440       by blast
```
```  1441     then have "?xs \<in> ?A"  using nxs  by blast}
```
```  1442   ultimately show ?thesis by auto
```
```  1443 qed
```
```  1444
```
```  1445     (* The general form *)
```
```  1446 lemma fps_setprod_nth:
```
```  1447   fixes m :: nat and a :: "nat \<Rightarrow> ('a::comm_ring_1) fps"
```
```  1448   shows "(setprod a {0 .. m})\$n = setsum (\<lambda>v. setprod (\<lambda>j. (a j) \$ (v!j)) {0..m}) (natpermute n (m+1))"
```
```  1449   (is "?P m n")
```
```  1450 proof(induct m arbitrary: n rule: nat_less_induct)
```
```  1451   fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"
```
```  1452   {assume m0: "m = 0"
```
```  1453     hence "?P m n" apply simp
```
```  1454       unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] by simp}
```
```  1455   moreover
```
```  1456   {fix k assume k: "m = Suc k"
```
```  1457     have km: "k < m" using k by arith
```
```  1458     have u0: "{0 .. k} \<union> {m} = {0..m}" using k apply (simp add: set_eq_iff) by presburger
```
```  1459     have f0: "finite {0 .. k}" "finite {m}" by auto
```
```  1460     have d0: "{0 .. k} \<inter> {m} = {}" using k by auto
```
```  1461     have "(setprod a {0 .. m}) \$ n = (setprod a {0 .. k} * a m) \$ n"
```
```  1462       unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0] by simp
```
```  1463     also have "\<dots> = (\<Sum>i = 0..n. (\<Sum>v\<in>natpermute i (k + 1). \<Prod>j\<in>{0..k}. a j \$ v ! j) * a m \$ (n - i))"
```
```  1464       unfolding fps_mult_nth H[rule_format, OF km] ..
```
```  1465     also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j \$ v ! j)"
```
```  1466       apply (simp add: k)
```
```  1467       unfolding natpermute_split[of m "m + 1", simplified, of n, unfolded natlist_trivial_1[unfolded One_nat_def] k]
```
```  1468       apply (subst setsum_UN_disjoint)
```
```  1469       apply simp
```
```  1470       apply simp
```
```  1471       unfolding image_Collect[symmetric]
```
```  1472       apply clarsimp
```
```  1473       apply (rule finite_imageI)
```
```  1474       apply (rule natpermute_finite)
```
```  1475       apply (clarsimp simp add: set_eq_iff)
```
```  1476       apply auto
```
```  1477       apply (rule setsum_cong2)
```
```  1478       unfolding setsum_left_distrib
```
```  1479       apply (rule sym)
```
```  1480       apply (rule_tac f="\<lambda>xs. xs @[n - x]" in  setsum_reindex_cong)
```
```  1481       apply (simp add: inj_on_def)
```
```  1482       apply auto
```
```  1483       unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded k]
```
```  1484       apply (clarsimp simp add: natpermute_def nth_append)
```
```  1485       done
```
```  1486     finally have "?P m n" .}
```
```  1487   ultimately show "?P m n " by (cases m, auto)
```
```  1488 qed
```
```  1489
```
```  1490 text{* The special form for powers *}
```
```  1491 lemma fps_power_nth_Suc:
```
```  1492   fixes m :: nat and a :: "('a::comm_ring_1) fps"
```
```  1493   shows "(a ^ Suc m)\$n = setsum (\<lambda>v. setprod (\<lambda>j. a \$ (v!j)) {0..m}) (natpermute n (m+1))"
```
```  1494 proof-
```
```  1495   have f: "finite {0 ..m}" by simp
```
```  1496   have th0: "a^Suc m = setprod (\<lambda>i. a) {0..m}" unfolding setprod_constant[OF f, of a] by simp
```
```  1497   show ?thesis unfolding th0 fps_setprod_nth ..
```
```  1498 qed
```
```  1499 lemma fps_power_nth:
```
```  1500   fixes m :: nat and a :: "('a::comm_ring_1) fps"
```
```  1501   shows "(a ^m)\$n = (if m=0 then 1\$n else setsum (\<lambda>v. setprod (\<lambda>j. a \$ (v!j)) {0..m - 1}) (natpermute n m))"
```
```  1502   by (cases m, simp_all add: fps_power_nth_Suc del: power_Suc)
```
```  1503
```
```  1504 lemma fps_nth_power_0:
```
```  1505   fixes m :: nat and a :: "('a::{comm_ring_1}) fps"
```
```  1506   shows "(a ^m)\$0 = (a\$0) ^ m"
```
```  1507 proof-
```
```  1508   {assume "m=0" hence ?thesis by simp}
```
```  1509   moreover
```
```  1510   {fix n assume m: "m = Suc n"
```
```  1511     have c: "m = card {0..n}" using m by simp
```
```  1512    have "(a ^m)\$0 = setprod (\<lambda>i. a\$0) {0..n}"
```
```  1513      by (simp add: m fps_power_nth del: replicate.simps power_Suc)
```
```  1514    also have "\<dots> = (a\$0) ^ m"
```
```  1515      unfolding c by (rule setprod_constant, simp)
```
```  1516    finally have ?thesis .}
```
```  1517  ultimately show ?thesis by (cases m, auto)
```
```  1518 qed
```
```  1519
```
```  1520 lemma fps_compose_inj_right:
```
```  1521   assumes a0: "a\$0 = (0::'a::{idom})"
```
```  1522   and a1: "a\$1 \<noteq> 0"
```
```  1523   shows "(b oo a = c oo a) \<longleftrightarrow> b = c" (is "?lhs \<longleftrightarrow>?rhs")
```
```  1524 proof-
```
```  1525   {assume ?rhs then have "?lhs" by simp}
```
```  1526   moreover
```
```  1527   {assume h: ?lhs
```
```  1528     {fix n have "b\$n = c\$n"
```
```  1529       proof(induct n rule: nat_less_induct)
```
```  1530         fix n assume H: "\<forall>m<n. b\$m = c\$m"
```
```  1531         {assume n0: "n=0"
```
```  1532           from h have "(b oo a)\$n = (c oo a)\$n" by simp
```
```  1533           hence "b\$n = c\$n" using n0 by (simp add: fps_compose_nth)}
```
```  1534         moreover
```
```  1535         {fix n1 assume n1: "n = Suc n1"
```
```  1536           have f: "finite {0 .. n1}" "finite {n}" by simp_all
```
```  1537           have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto
```
```  1538           have d: "{0 .. n1} \<inter> {n} = {}" using n1 by auto
```
```  1539           have seq: "(\<Sum>i = 0..n1. b \$ i * a ^ i \$ n) = (\<Sum>i = 0..n1. c \$ i * a ^ i \$ n)"
```
```  1540             apply (rule setsum_cong2)
```
```  1541             using H n1 by auto
```
```  1542           have th0: "(b oo a) \$n = (\<Sum>i = 0..n1. c \$ i * a ^ i \$ n) + b\$n * (a\$1)^n"
```
```  1543             unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq] seq
```
```  1544             using startsby_zero_power_nth_same[OF a0]
```
```  1545             by simp
```
```  1546           have th1: "(c oo a) \$n = (\<Sum>i = 0..n1. c \$ i * a ^ i \$ n) + c\$n * (a\$1)^n"
```
```  1547             unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq]
```
```  1548             using startsby_zero_power_nth_same[OF a0]
```
```  1549             by simp
```
```  1550           from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
```
```  1551           have "b\$n = c\$n" by auto}
```
```  1552         ultimately show "b\$n = c\$n" by (cases n, auto)
```
```  1553       qed}
```
```  1554     then have ?rhs by (simp add: fps_eq_iff)}
```
```  1555   ultimately show ?thesis by blast
```
```  1556 qed
```
```  1557
```
```  1558
```
```  1559 subsection {* Radicals *}
```
```  1560
```
```  1561 declare setprod_cong[fundef_cong]
```
```  1562 function radical :: "(nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> ('a::{field}) fps \<Rightarrow> nat \<Rightarrow> 'a" where
```
```  1563   "radical r 0 a 0 = 1"
```
```  1564 | "radical r 0 a (Suc n) = 0"
```
```  1565 | "radical r (Suc k) a 0 = r (Suc k) (a\$0)"
```
```  1566 | "radical r (Suc k) a (Suc n) = (a\$ Suc n - setsum (\<lambda>xs. setprod (\<lambda>j. radical r (Suc k) a (xs ! j)) {0..k}) {xs. xs \<in> natpermute (Suc n) (Suc k) \<and> Suc n \<notin> set xs}) / (of_nat (Suc k) * (radical r (Suc k) a 0)^k)"
```
```  1567 by pat_completeness auto
```
```  1568
```
```  1569 termination radical
```
```  1570 proof
```
```  1571   let ?R = "measure (\<lambda>(r, k, a, n). n)"
```
```  1572   {
```
```  1573     show "wf ?R" by auto}
```
```  1574   {fix r k a n xs i
```
```  1575     assume xs: "xs \<in> {xs \<in> natpermute (Suc n) (Suc k). Suc n \<notin> set xs}" and i: "i \<in> {0..k}"
```
```  1576     {assume c: "Suc n \<le> xs ! i"
```
```  1577       from xs i have "xs !i \<noteq> Suc n" by (auto simp add: in_set_conv_nth natpermute_def)
```
```  1578       with c have c': "Suc n < xs!i" by arith
```
```  1579       have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all
```
```  1580       have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" by auto
```
```  1581       have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" using i by auto
```
```  1582       from xs have "Suc n = foldl op + 0 xs" by (simp add: natpermute_def)
```
```  1583       also have "\<dots> = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs
```
```  1584         by (simp add: natpermute_def)
```
```  1585       also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
```
```  1586         unfolding eqs  setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
```
```  1587         unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
```
```  1588         by simp
```
```  1589       finally have False using c' by simp}
```
```  1590     then show "((r,Suc k,a,xs!i), r, Suc k, a, Suc n) \<in> ?R"
```
```  1591       apply auto by (metis not_less)}
```
```  1592   {fix r k a n
```
```  1593     show "((r,Suc k, a, 0),r, Suc k, a, Suc n) \<in> ?R" by simp}
```
```  1594 qed
```
```  1595
```
```  1596 definition "fps_radical r n a = Abs_fps (radical r n a)"
```
```  1597
```
```  1598 lemma fps_radical0[simp]: "fps_radical r 0 a = 1"
```
```  1599   apply (auto simp add: fps_eq_iff fps_radical_def)  by (case_tac n, auto)
```
```  1600
```
```  1601 lemma fps_radical_nth_0[simp]: "fps_radical r n a \$ 0 = (if n=0 then 1 else r n (a\$0))"
```
```  1602   by (cases n, simp_all add: fps_radical_def)
```
```  1603
```
```  1604 lemma fps_radical_power_nth[simp]:
```
```  1605   assumes r: "(r k (a\$0)) ^ k = a\$0"
```
```  1606   shows "fps_radical r k a ^ k \$ 0 = (if k = 0 then 1 else a\$0)"
```
```  1607 proof-
```
```  1608   {assume "k=0" hence ?thesis by simp }
```
```  1609   moreover
```
```  1610   {fix h assume h: "k = Suc h"
```
```  1611     have fh: "finite {0..h}" by simp
```
```  1612     have eq1: "fps_radical r k a ^ k \$ 0 = (\<Prod>j\<in>{0..h}. fps_radical r k a \$ (replicate k 0) ! j)"
```
```  1613       unfolding fps_power_nth h by simp
```
```  1614     also have "\<dots> = (\<Prod>j\<in>{0..h}. r k (a\$0))"
```
```  1615       apply (rule setprod_cong)
```
```  1616       apply simp
```
```  1617       using h
```
```  1618       apply (subgoal_tac "replicate k (0::nat) ! x = 0")
```
```  1619       by (auto intro: nth_replicate simp del: replicate.simps)
```
```  1620     also have "\<dots> = a\$0"
```
```  1621       unfolding setprod_constant[OF fh] using r by (simp add: h)
```
```  1622     finally have ?thesis using h by simp}
```
```  1623   ultimately show ?thesis by (cases k, auto)
```
```  1624 qed
```
```  1625
```
```  1626 lemma natpermute_max_card: assumes n0: "n\<noteq>0"
```
```  1627   shows "card {xs \<in> natpermute n (k+1). n \<in> set xs} = k+1"
```
```  1628   unfolding natpermute_contain_maximal
```
```  1629 proof-
```
```  1630   let ?A= "\<lambda>i. {replicate (k + 1) 0[i := n]}"
```
```  1631   let ?K = "{0 ..k}"
```
```  1632   have fK: "finite ?K" by simp
```
```  1633   have fAK: "\<forall>i\<in>?K. finite (?A i)" by auto
```
```  1634   have d: "\<forall>i\<in> ?K. \<forall>j\<in> ?K. i \<noteq> j \<longrightarrow> {replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
```
```  1635   proof(clarify)
```
```  1636     fix i j assume i: "i \<in> ?K" and j: "j\<in> ?K" and ij: "i\<noteq>j"
```
```  1637     {assume eq: "replicate (k+1) 0 [i:=n] = replicate (k+1) 0 [j:= n]"
```
```  1638       have "(replicate (k+1) 0 [i:=n] ! i) = n" using i by (simp del: replicate.simps)
```
```  1639       moreover
```
```  1640       have "(replicate (k+1) 0 [j:=n] ! i) = 0" using i ij by (simp del: replicate.simps)
```
```  1641       ultimately have False using eq n0 by (simp del: replicate.simps)}
```
```  1642     then show "{replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
```
```  1643       by auto
```
```  1644   qed
```
```  1645   from card_UN_disjoint[OF fK fAK d]
```
```  1646   show "card (\<Union>i\<in>{0..k}. {replicate (k + 1) 0[i := n]}) = k+1" by simp
```
```  1647 qed
```
```  1648
```
```  1649 lemma power_radical:
```
```  1650   fixes a:: "'a::field_char_0 fps"
```
```  1651   assumes a0: "a\$0 \<noteq> 0"
```
```  1652   shows "(r (Suc k) (a\$0)) ^ Suc k = a\$0 \<longleftrightarrow> (fps_radical r (Suc k) a) ^ (Suc k) = a"
```
```  1653 proof-
```
```  1654   let ?r = "fps_radical r (Suc k) a"
```
```  1655   {assume r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0"
```
```  1656     from a0 r0 have r00: "r (Suc k) (a\$0) \<noteq> 0" by auto
```
```  1657     {fix z have "?r ^ Suc k \$ z = a\$z"
```
```  1658       proof(induct z rule: nat_less_induct)
```
```  1659         fix n assume H: "\<forall>m<n. ?r ^ Suc k \$ m = a\$m"
```
```  1660         {assume "n = 0" hence "?r ^ Suc k \$ n = a \$n"
```
```  1661             using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp}
```
```  1662         moreover
```
```  1663         {fix n1 assume n1: "n = Suc n1"
```
```  1664           have fK: "finite {0..k}" by simp
```
```  1665           have nz: "n \<noteq> 0" using n1 by arith
```
```  1666           let ?Pnk = "natpermute n (k + 1)"
```
```  1667           let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
```
```  1668           let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
```
```  1669           have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
```
```  1670           have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
```
```  1671           have f: "finite ?Pnkn" "finite ?Pnknn"
```
```  1672             using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
```
```  1673             by (metis natpermute_finite)+
```
```  1674           let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r \$ v ! j"
```
```  1675           have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r \$ n * r (Suc k) (a \$ 0) ^ k) ?Pnkn"
```
```  1676           proof(rule setsum_cong2)
```
```  1677             fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
```
```  1678             let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a \$ v ! j) = fps_radical r (Suc k) a \$ n * r (Suc k) (a \$ 0) ^ k"
```
```  1679           from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
```
```  1680             unfolding natpermute_contain_maximal by auto
```
```  1681           have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a \$ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a \$ n else r (Suc k) (a\$0))"
```
```  1682             apply (rule setprod_cong, simp)
```
```  1683             using i r0 by (simp del: replicate.simps)
```
```  1684           also have "\<dots> = (fps_radical r (Suc k) a \$ n) * r (Suc k) (a\$0) ^ k"
```
```  1685             unfolding setprod_gen_delta[OF fK] using i r0 by simp
```
```  1686           finally show ?ths .
```
```  1687         qed
```
```  1688         then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r \$ n * r (Suc k) (a \$ 0) ^ k"
```
```  1689           by (simp add: natpermute_max_card[OF nz, simplified])
```
```  1690         also have "\<dots> = a\$n - setsum ?f ?Pnknn"
```
```  1691           unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc)
```
```  1692         finally have fn: "setsum ?f ?Pnkn = a\$n - setsum ?f ?Pnknn" .
```
```  1693         have "(?r ^ Suc k)\$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
```
```  1694           unfolding fps_power_nth_Suc setsum_Un_disjoint[OF f d, unfolded eq] ..
```
```  1695         also have "\<dots> = a\$n" unfolding fn by simp
```
```  1696         finally have "?r ^ Suc k \$ n = a \$n" .}
```
```  1697       ultimately  show "?r ^ Suc k \$ n = a \$n" by (cases n, auto)
```
```  1698     qed }
```
```  1699   then have ?thesis using r0 by (simp add: fps_eq_iff)}
```
```  1700 moreover
```
```  1701 { assume h: "(fps_radical r (Suc k) a) ^ (Suc k) = a"
```
```  1702   hence "((fps_radical r (Suc k) a) ^ (Suc k))\$0 = a\$0" by simp
```
```  1703   then have "(r (Suc k) (a\$0)) ^ Suc k = a\$0"
```
```  1704     unfolding fps_power_nth_Suc
```
```  1705     by (simp add: setprod_constant del: replicate.simps)}
```
```  1706 ultimately show ?thesis by blast
```
```  1707 qed
```
```  1708
```
```  1709 (*
```
```  1710 lemma power_radical:
```
```  1711   fixes a:: "'a::field_char_0 fps"
```
```  1712   assumes r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0" and a0: "a\$0 \<noteq> 0"
```
```  1713   shows "(fps_radical r (Suc k) a) ^ (Suc k) = a"
```
```  1714 proof-
```
```  1715   let ?r = "fps_radical r (Suc k) a"
```
```  1716   from a0 r0 have r00: "r (Suc k) (a\$0) \<noteq> 0" by auto
```
```  1717   {fix z have "?r ^ Suc k \$ z = a\$z"
```
```  1718     proof(induct z rule: nat_less_induct)
```
```  1719       fix n assume H: "\<forall>m<n. ?r ^ Suc k \$ m = a\$m"
```
```  1720       {assume "n = 0" hence "?r ^ Suc k \$ n = a \$n"
```
```  1721           using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp}
```
```  1722       moreover
```
```  1723       {fix n1 assume n1: "n = Suc n1"
```
```  1724         have fK: "finite {0..k}" by simp
```
```  1725         have nz: "n \<noteq> 0" using n1 by arith
```
```  1726         let ?Pnk = "natpermute n (k + 1)"
```
```  1727         let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
```
```  1728         let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
```
```  1729         have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
```
```  1730         have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
```
```  1731         have f: "finite ?Pnkn" "finite ?Pnknn"
```
```  1732           using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
```
```  1733           by (metis natpermute_finite)+
```
```  1734         let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r \$ v ! j"
```
```  1735         have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r \$ n * r (Suc k) (a \$ 0) ^ k) ?Pnkn"
```
```  1736         proof(rule setsum_cong2)
```
```  1737           fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
```
```  1738           let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a \$ v ! j) = fps_radical r (Suc k) a \$ n * r (Suc k) (a \$ 0) ^ k"
```
```  1739           from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
```
```  1740             unfolding natpermute_contain_maximal by auto
```
```  1741           have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a \$ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a \$ n else r (Suc k) (a\$0))"
```
```  1742             apply (rule setprod_cong, simp)
```
```  1743             using i r0 by (simp del: replicate.simps)
```
```  1744           also have "\<dots> = (fps_radical r (Suc k) a \$ n) * r (Suc k) (a\$0) ^ k"
```
```  1745             unfolding setprod_gen_delta[OF fK] using i r0 by simp
```
```  1746           finally show ?ths .
```
```  1747         qed
```
```  1748         then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r \$ n * r (Suc k) (a \$ 0) ^ k"
```
```  1749           by (simp add: natpermute_max_card[OF nz, simplified])
```
```  1750         also have "\<dots> = a\$n - setsum ?f ?Pnknn"
```
```  1751           unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc )
```
```  1752         finally have fn: "setsum ?f ?Pnkn = a\$n - setsum ?f ?Pnknn" .
```
```  1753         have "(?r ^ Suc k)\$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
```
```  1754           unfolding fps_power_nth_Suc setsum_Un_disjoint[OF f d, unfolded eq] ..
```
```  1755         also have "\<dots> = a\$n" unfolding fn by simp
```
```  1756         finally have "?r ^ Suc k \$ n = a \$n" .}
```
```  1757       ultimately  show "?r ^ Suc k \$ n = a \$n" by (cases n, auto)
```
```  1758   qed }
```
```  1759   then show ?thesis by (simp add: fps_eq_iff)
```
```  1760 qed
```
```  1761
```
```  1762 *)
```
```  1763 lemma eq_divide_imp': assumes c0: "(c::'a::field) ~= 0" and eq: "a * c = b"
```
```  1764   shows "a = b / c"
```
```  1765 proof-
```
```  1766   from eq have "a * c * inverse c = b * inverse c" by simp
```
```  1767   hence "a * (inverse c * c) = b/c" by (simp only: field_simps divide_inverse)
```
```  1768   then show "a = b/c" unfolding  field_inverse[OF c0] by simp
```
```  1769 qed
```
```  1770
```
```  1771 lemma radical_unique:
```
```  1772   assumes r0: "(r (Suc k) (b\$0)) ^ Suc k = b\$0"
```
```  1773   and a0: "r (Suc k) (b\$0 ::'a::field_char_0) = a\$0" and b0: "b\$0 \<noteq> 0"
```
```  1774   shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b"
```
```  1775 proof-
```
```  1776   let ?r = "fps_radical r (Suc k) b"
```
```  1777   have r00: "r (Suc k) (b\$0) \<noteq> 0" using b0 r0 by auto
```
```  1778   {assume H: "a = ?r"
```
```  1779     from H have "a^Suc k = b" using power_radical[OF b0, of r k, unfolded r0] by simp}
```
```  1780   moreover
```
```  1781   {assume H: "a^Suc k = b"
```
```  1782     have ceq: "card {0..k} = Suc k" by simp
```
```  1783     have fk: "finite {0..k}" by simp
```
```  1784     from a0 have a0r0: "a\$0 = ?r\$0" by simp
```
```  1785     {fix n have "a \$ n = ?r \$ n"
```
```  1786       proof(induct n rule: nat_less_induct)
```
```  1787         fix n assume h: "\<forall>m<n. a\$m = ?r \$m"
```
```  1788         {assume "n = 0" hence "a\$n = ?r \$n" using a0 by simp }
```
```  1789         moreover
```
```  1790         {fix n1 assume n1: "n = Suc n1"
```
```  1791           have fK: "finite {0..k}" by simp
```
```  1792         have nz: "n \<noteq> 0" using n1 by arith
```
```  1793         let ?Pnk = "natpermute n (Suc k)"
```
```  1794         let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
```
```  1795         let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
```
```  1796         have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
```
```  1797         have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
```
```  1798         have f: "finite ?Pnkn" "finite ?Pnknn"
```
```  1799           using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
```
```  1800           by (metis natpermute_finite)+
```
```  1801         let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r \$ v ! j"
```
```  1802         let ?g = "\<lambda>v. \<Prod>j\<in>{0..k}. a \$ v ! j"
```
```  1803         have "setsum ?g ?Pnkn = setsum (\<lambda>v. a \$ n * (?r\$0)^k) ?Pnkn"
```
```  1804         proof(rule setsum_cong2)
```
```  1805           fix v assume v: "v \<in> {xs \<in> natpermute n (Suc k). n \<in> set xs}"
```
```  1806           let ?ths = "(\<Prod>j\<in>{0..k}. a \$ v ! j) = a \$ n * (?r\$0)^k"
```
```  1807           from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
```
```  1808             unfolding Suc_eq_plus1 natpermute_contain_maximal by (auto simp del: replicate.simps)
```
```  1809           have "(\<Prod>j\<in>{0..k}. a \$ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then a \$ n else r (Suc k) (b\$0))"
```
```  1810             apply (rule setprod_cong, simp)
```
```  1811             using i a0 by (simp del: replicate.simps)
```
```  1812           also have "\<dots> = a \$ n * (?r \$ 0)^k"
```
```  1813             unfolding  setprod_gen_delta[OF fK] using i by simp
```
```  1814           finally show ?ths .
```
```  1815         qed
```
```  1816         then have th0: "setsum ?g ?Pnkn = of_nat (k+1) * a \$ n * (?r \$ 0)^k"
```
```  1817           by (simp add: natpermute_max_card[OF nz, simplified])
```
```  1818         have th1: "setsum ?g ?Pnknn = setsum ?f ?Pnknn"
```
```  1819         proof (rule setsum_cong2, rule setprod_cong, simp)
```
```  1820           fix xs i assume xs: "xs \<in> ?Pnknn" and i: "i \<in> {0..k}"
```
```  1821           {assume c: "n \<le> xs ! i"
```
```  1822             from xs i have "xs !i \<noteq> n" by (auto simp add: in_set_conv_nth natpermute_def)
```
```  1823             with c have c': "n < xs!i" by arith
```
```  1824             have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all
```
```  1825             have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" by auto
```
```  1826             have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" using i by auto
```
```  1827             from xs have "n = foldl op + 0 xs" by (simp add: natpermute_def)
```
```  1828             also have "\<dots> = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs
```
```  1829               by (simp add: natpermute_def)
```
```  1830             also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
```
```  1831               unfolding eqs  setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
```
```  1832               unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
```
```  1833               by simp
```
```  1834             finally have False using c' by simp}
```
```  1835           then have thn: "xs!i < n" by arith
```
```  1836           from h[rule_format, OF thn]
```
```  1837           show "a\$(xs !i) = ?r\$(xs!i)" .
```
```  1838         qed
```
```  1839         have th00: "\<And>(x::'a). of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x"
```
```  1840           by (simp add: field_simps del: of_nat_Suc)
```
```  1841         from H have "b\$n = a^Suc k \$ n" by (simp add: fps_eq_iff)
```
```  1842         also have "a ^ Suc k\$n = setsum ?g ?Pnkn + setsum ?g ?Pnknn"
```
```  1843           unfolding fps_power_nth_Suc
```
```  1844           using setsum_Un_disjoint[OF f d, unfolded Suc_eq_plus1[symmetric],
```
```  1845             unfolded eq, of ?g] by simp
```
```  1846         also have "\<dots> = of_nat (k+1) * a \$ n * (?r \$ 0)^k + setsum ?f ?Pnknn" unfolding th0 th1 ..
```
```  1847         finally have "of_nat (k+1) * a \$ n * (?r \$ 0)^k = b\$n - setsum ?f ?Pnknn" by simp
```
```  1848         then have "a\$n = (b\$n - setsum ?f ?Pnknn) / (of_nat (k+1) * (?r \$ 0)^k)"
```
```  1849           apply -
```
```  1850           apply (rule eq_divide_imp')
```
```  1851           using r00
```
```  1852           apply (simp del: of_nat_Suc)
```
```  1853           by (simp add: mult_ac)
```
```  1854         then have "a\$n = ?r \$n"
```
```  1855           apply (simp del: of_nat_Suc)
```
```  1856           unfolding fps_radical_def n1
```
```  1857           by (simp add: field_simps n1 th00 del: of_nat_Suc)}
```
```  1858         ultimately show "a\$n = ?r \$ n" by (cases n, auto)
```
```  1859       qed}
```
```  1860     then have "a = ?r" by (simp add: fps_eq_iff)}
```
```  1861   ultimately show ?thesis by blast
```
```  1862 qed
```
```  1863
```
```  1864
```
```  1865 lemma radical_power:
```
```  1866   assumes r0: "r (Suc k) ((a\$0) ^ Suc k) = a\$0"
```
```  1867   and a0: "(a\$0 ::'a::field_char_0) \<noteq> 0"
```
```  1868   shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
```
```  1869 proof-
```
```  1870   let ?ak = "a^ Suc k"
```
```  1871   have ak0: "?ak \$ 0 = (a\$0) ^ Suc k" by (simp add: fps_nth_power_0 del: power_Suc)
```
```  1872   from r0 have th0: "r (Suc k) (a ^ Suc k \$ 0) ^ Suc k = a ^ Suc k \$ 0" using ak0 by auto
```
```  1873   from r0 ak0 have th1: "r (Suc k) (a ^ Suc k \$ 0) = a \$ 0" by auto
```
```  1874   from ak0 a0 have ak00: "?ak \$ 0 \<noteq>0 " by auto
```
```  1875   from radical_unique[of r k ?ak a, OF th0 th1 ak00] show ?thesis by metis
```
```  1876 qed
```
```  1877
```
```  1878 lemma fps_deriv_radical:
```
```  1879   fixes a:: "'a::field_char_0 fps"
```
```  1880   assumes r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0" and a0: "a\$0 \<noteq> 0"
```
```  1881   shows "fps_deriv (fps_radical r (Suc k) a) = fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
```
```  1882 proof-
```
```  1883   let ?r= "fps_radical r (Suc k) a"
```
```  1884   let ?w = "(fps_const (of_nat (Suc k)) * ?r ^ k)"
```
```  1885   from a0 r0 have r0': "r (Suc k) (a\$0) \<noteq> 0" by auto
```
```  1886   from r0' have w0: "?w \$ 0 \<noteq> 0" by (simp del: of_nat_Suc)
```
```  1887   note th0 = inverse_mult_eq_1[OF w0]
```
```  1888   let ?iw = "inverse ?w"
```
```  1889   from iffD1[OF power_radical[of a r], OF a0 r0]
```
```  1890   have "fps_deriv (?r ^ Suc k) = fps_deriv a" by simp
```
```  1891   hence "fps_deriv ?r * ?w = fps_deriv a"
```
```  1892     by (simp add: fps_deriv_power mult_ac del: power_Suc)
```
```  1893   hence "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a" by simp
```
```  1894   hence "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w"
```
```  1895     by (simp add: fps_divide_def)
```
```  1896   then show ?thesis unfolding th0 by simp
```
```  1897 qed
```
```  1898
```
```  1899 lemma radical_mult_distrib:
```
```  1900   fixes a:: "'a::field_char_0 fps"
```
```  1901   assumes
```
```  1902   k: "k > 0"
```
```  1903   and ra0: "r k (a \$ 0) ^ k = a \$ 0"
```
```  1904   and rb0: "r k (b \$ 0) ^ k = b \$ 0"
```
```  1905   and a0: "a\$0 \<noteq> 0"
```
```  1906   and b0: "b\$0 \<noteq> 0"
```
```  1907   shows "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0) \<longleftrightarrow> fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
```
```  1908 proof-
```
```  1909   {assume  r0': "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0)"
```
```  1910   from r0' have r0: "(r (k) ((a*b)\$0)) ^ k = (a*b)\$0"
```
```  1911     by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
```
```  1912   {assume "k=0" hence ?thesis using r0' by simp}
```
```  1913   moreover
```
```  1914   {fix h assume k: "k = Suc h"
```
```  1915   let ?ra = "fps_radical r (Suc h) a"
```
```  1916   let ?rb = "fps_radical r (Suc h) b"
```
```  1917   have th0: "r (Suc h) ((a * b) \$ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) \$ 0"
```
```  1918     using r0' k by (simp add: fps_mult_nth)
```
```  1919   have ab0: "(a*b) \$ 0 \<noteq> 0" using a0 b0 by (simp add: fps_mult_nth)
```
```  1920   from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric]
```
```  1921     iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded k]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded k]] k r0'
```
```  1922   have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)}
```
```  1923 ultimately have ?thesis by (cases k, auto)}
```
```  1924 moreover
```
```  1925 {assume h: "fps_radical r k (a*b) = fps_radical r k a * fps_radical r k b"
```
```  1926   hence "(fps_radical r k (a*b))\$0 = (fps_radical r k a * fps_radical r k b)\$0" by simp
```
```  1927   then have "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0)"
```
```  1928     using k by (simp add: fps_mult_nth)}
```
```  1929 ultimately show ?thesis by blast
```
```  1930 qed
```
```  1931
```
```  1932 (*
```
```  1933 lemma radical_mult_distrib:
```
```  1934   fixes a:: "'a::field_char_0 fps"
```
```  1935   assumes
```
```  1936   ra0: "r k (a \$ 0) ^ k = a \$ 0"
```
```  1937   and rb0: "r k (b \$ 0) ^ k = b \$ 0"
```
```  1938   and r0': "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0)"
```
```  1939   and a0: "a\$0 \<noteq> 0"
```
```  1940   and b0: "b\$0 \<noteq> 0"
```
```  1941   shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
```
```  1942 proof-
```
```  1943   from r0' have r0: "(r (k) ((a*b)\$0)) ^ k = (a*b)\$0"
```
```  1944     by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
```
```  1945   {assume "k=0" hence ?thesis by simp}
```
```  1946   moreover
```
```  1947   {fix h assume k: "k = Suc h"
```
```  1948   let ?ra = "fps_radical r (Suc h) a"
```
```  1949   let ?rb = "fps_radical r (Suc h) b"
```
```  1950   have th0: "r (Suc h) ((a * b) \$ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) \$ 0"
```
```  1951     using r0' k by (simp add: fps_mult_nth)
```
```  1952   have ab0: "(a*b) \$ 0 \<noteq> 0" using a0 b0 by (simp add: fps_mult_nth)
```
```  1953   from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric]
```
```  1954     power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k
```
```  1955   have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)}
```
```  1956 ultimately show ?thesis by (cases k, auto)
```
```  1957 qed
```
```  1958 *)
```
```  1959
```
```  1960 lemma fps_divide_1[simp]: "(a:: ('a::field) fps) / 1 = a"
```
```  1961   by (simp add: fps_divide_def)
```
```  1962
```
```  1963 lemma radical_divide:
```
```  1964   fixes a :: "'a::field_char_0 fps"
```
```  1965   assumes
```
```  1966   kp: "k>0"
```
```  1967   and ra0: "(r k (a \$ 0)) ^ k = a \$ 0"
```
```  1968   and rb0: "(r k (b \$ 0)) ^ k = b \$ 0"
```
```  1969   and a0: "a\$0 \<noteq> 0"
```
```  1970   and b0: "b\$0 \<noteq> 0"
```
```  1971   shows "r k ((a \$ 0) / (b\$0)) = r k (a\$0) / r k (b \$ 0) \<longleftrightarrow> fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b" (is "?lhs = ?rhs")
```
```  1972 proof-
```
```  1973   let ?r = "fps_radical r k"
```
```  1974   from kp obtain h where k: "k = Suc h" by (cases k, auto)
```
```  1975   have ra0': "r k (a\$0) \<noteq> 0" using a0 ra0 k by auto
```
```  1976   have rb0': "r k (b\$0) \<noteq> 0" using b0 rb0 k by auto
```
```  1977
```
```  1978   {assume ?rhs
```
```  1979     then have "?r (a/b) \$ 0 = (?r a / ?r b)\$0" by simp
```
```  1980     then have ?lhs using k a0 b0 rb0'
```
```  1981       by (simp add: fps_divide_def fps_mult_nth fps_inverse_def divide_inverse) }
```
```  1982   moreover
```
```  1983   {assume h: ?lhs
```
```  1984     from a0 b0 have ab0[simp]: "(a/b)\$0 = a\$0 / b\$0"
```
```  1985       by (simp add: fps_divide_def fps_mult_nth divide_inverse fps_inverse_def)
```
```  1986     have th0: "r k ((a/b)\$0) ^ k = (a/b)\$0"
```
```  1987       by (simp add: h nonzero_power_divide[OF rb0'] ra0 rb0 del: k)
```
```  1988     from a0 b0 ra0' rb0' kp h
```
```  1989     have th1: "r k ((a / b) \$ 0) = (fps_radical r k a / fps_radical r k b) \$ 0"
```
```  1990       by (simp add: fps_divide_def fps_mult_nth fps_inverse_def divide_inverse del: k)
```
```  1991     from a0 b0 ra0' rb0' kp have ab0': "(a / b) \$ 0 \<noteq> 0"
```
```  1992       by (simp add: fps_divide_def fps_mult_nth fps_inverse_def nonzero_imp_inverse_nonzero)
```
```  1993     note tha[simp] = iffD1[OF power_radical[where r=r and k=h], OF a0 ra0[unfolded k], unfolded k[symmetric]]
```
```  1994     note thb[simp] = iffD1[OF power_radical[where r=r and k=h], OF b0 rb0[unfolded k], unfolded k[symmetric]]
```
```  1995     have th2: "(?r a / ?r b)^k = a/b"
```
```  1996       by (simp add: fps_divide_def power_mult_distrib fps_inverse_power[symmetric])
```
```  1997     from iffD1[OF radical_unique[where r=r and a="?r a / ?r b" and b="a/b" and k=h], symmetric, unfolded k[symmetric], OF th0 th1 ab0' th2] have ?rhs .}
```
```  1998   ultimately show ?thesis by blast
```
```  1999 qed
```
```  2000
```
```  2001 lemma radical_inverse:
```
```  2002   fixes a :: "'a::field_char_0 fps"
```
```  2003   assumes
```
```  2004   k: "k>0"
```
```  2005   and ra0: "r k (a \$ 0) ^ k = a \$ 0"
```
```  2006   and r1: "(r k 1)^k = 1"
```
```  2007   and a0: "a\$0 \<noteq> 0"
```
```  2008   shows "r k (inverse (a \$ 0)) = r k 1 / (r k (a \$ 0)) \<longleftrightarrow> fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a"
```
```  2009   using radical_divide[where k=k and r=r and a=1 and b=a, OF k ] ra0 r1 a0
```
```  2010   by (simp add: divide_inverse fps_divide_def)
```
```  2011
```
```  2012 subsection{* Derivative of composition *}
```
```  2013
```
```  2014 lemma fps_compose_deriv:
```
```  2015   fixes a:: "('a::idom) fps"
```
```  2016   assumes b0: "b\$0 = 0"
```
```  2017   shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * (fps_deriv b)"
```
```  2018 proof-
```
```  2019   {fix n
```
```  2020     have "(fps_deriv (a oo b))\$n = setsum (\<lambda>i. a \$ i * (fps_deriv (b^i))\$n) {0.. Suc n}"
```
```  2021       by (simp add: fps_compose_def field_simps setsum_right_distrib del: of_nat_Suc)
```
```  2022     also have "\<dots> = setsum (\<lambda>i. a\$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))\$n) {0.. Suc n}"
```
```  2023       by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
```
```  2024   also have "\<dots> = setsum (\<lambda>i. of_nat i * a\$i * (((b^(i - 1)) * fps_deriv b))\$n) {0.. Suc n}"
```
```  2025     unfolding fps_mult_left_const_nth  by (simp add: field_simps)
```
```  2026   also have "\<dots> = setsum (\<lambda>i. of_nat i * a\$i * (setsum (\<lambda>j. (b^ (i - 1))\$j * (fps_deriv b)\$(n - j)) {0..n})) {0.. Suc n}"
```
```  2027     unfolding fps_mult_nth ..
```
```  2028   also have "\<dots> = setsum (\<lambda>i. of_nat i * a\$i * (setsum (\<lambda>j. (b^ (i - 1))\$j * (fps_deriv b)\$(n - j)) {0..n})) {1.. Suc n}"
```
```  2029     apply (rule setsum_mono_zero_right)
```
```  2030     apply (auto simp add: mult_delta_left setsum_delta not_le)
```
```  2031     done
```
```  2032   also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a\$(i+1) * (setsum (\<lambda>j. (b^ i)\$j * of_nat (n - j + 1) * b\$(n - j + 1)) {0..n})) {0.. n}"
```
```  2033     unfolding fps_deriv_nth
```
```  2034     apply (rule setsum_reindex_cong [where f = Suc])
```
```  2035     by (auto simp add: mult_assoc)
```
```  2036   finally have th0: "(fps_deriv (a oo b))\$n = setsum (\<lambda>i. of_nat (i + 1) * a\$(i+1) * (setsum (\<lambda>j. (b^ i)\$j * of_nat (n - j + 1) * b\$(n - j + 1)) {0..n})) {0.. n}" .
```
```  2037
```
```  2038   have "(((fps_deriv a) oo b) * (fps_deriv b))\$n = setsum (\<lambda>i. (fps_deriv b)\$ (n - i) * ((fps_deriv a) oo b)\$i) {0..n}"
```
```  2039     unfolding fps_mult_nth by (simp add: mult_ac)
```
```  2040   also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. of_nat (n - i +1) * b\$(n - i + 1) * of_nat (j + 1) * a\$(j+1) * (b^j)\$i) {0..n}) {0..n}"
```
```  2041     unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult_assoc
```
```  2042     apply (rule setsum_cong2)
```
```  2043     apply (rule setsum_mono_zero_left)
```
```  2044     apply (simp_all add: subset_eq)
```
```  2045     apply clarify
```
```  2046     apply (subgoal_tac "b^i\$x = 0")
```
```  2047     apply simp
```
```  2048     apply (rule startsby_zero_power_prefix[OF b0, rule_format])
```
```  2049     by simp
```
```  2050   also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a\$(i+1) * (setsum (\<lambda>j. (b^ i)\$j * of_nat (n - j + 1) * b\$(n - j + 1)) {0..n})) {0.. n}"
```
```  2051     unfolding setsum_right_distrib
```
```  2052     apply (subst setsum_commute)
```
```  2053     by ((rule setsum_cong2)+) simp
```
```  2054   finally have "(fps_deriv (a oo b))\$n = (((fps_deriv a) oo b) * (fps_deriv b)) \$n"
```
```  2055     unfolding th0 by simp}
```
```  2056 then show ?thesis by (simp add: fps_eq_iff)
```
```  2057 qed
```
```  2058
```
```  2059 lemma fps_mult_X_plus_1_nth:
```
```  2060   "((1+X)*a) \$n = (if n = 0 then (a\$n :: 'a::comm_ring_1) else a\$n + a\$(n - 1))"
```
```  2061 proof-
```
```  2062   {assume "n = 0" hence ?thesis by (simp add: fps_mult_nth )}
```
```  2063   moreover
```
```  2064   {fix m assume m: "n = Suc m"
```
```  2065     have "((1+X)*a) \$n = setsum (\<lambda>i. (1+X)\$i * a\$(n-i)) {0..n}"
```
```  2066       by (simp add: fps_mult_nth)
```
```  2067     also have "\<dots> = setsum (\<lambda>i. (1+X)\$i * a\$(n-i)) {0.. 1}"
```
```  2068       unfolding m
```
```  2069       apply (rule setsum_mono_zero_right)
```
```  2070       by (auto simp add: )
```
```  2071     also have "\<dots> = (if n = 0 then (a\$n :: 'a::comm_ring_1) else a\$n + a\$(n - 1))"
```
```  2072       unfolding m
```
```  2073       by (simp add: )
```
```  2074     finally have ?thesis .}
```
```  2075   ultimately show ?thesis by (cases n, auto)
```
```  2076 qed
```
```  2077
```
```  2078 subsection{* Finite FPS (i.e. polynomials) and X *}
```
```  2079 lemma fps_poly_sum_X:
```
```  2080   assumes z: "\<forall>i > n. a\$i = (0::'a::comm_ring_1)"
```
```  2081   shows "a = setsum (\<lambda>i. fps_const (a\$i) * X^i) {0..n}" (is "a = ?r")
```
```  2082 proof-
```
```  2083   {fix i
```
```  2084     have "a\$i = ?r\$i"
```
```  2085       unfolding fps_setsum_nth fps_mult_left_const_nth X_power_nth
```
```  2086       by (simp add: mult_delta_right setsum_delta' z)
```
```  2087   }
```
```  2088   then show ?thesis unfolding fps_eq_iff by blast
```
```  2089 qed
```
```  2090
```
```  2091 subsection{* Compositional inverses *}
```
```  2092
```
```  2093
```
```  2094 fun compinv :: "'a fps \<Rightarrow> nat \<Rightarrow> 'a::{field}" where
```
```  2095   "compinv a 0 = X\$0"
```
```  2096 | "compinv a (Suc n) = (X\$ Suc n - setsum (\<lambda>i. (compinv a i) * (a^i)\$Suc n) {0 .. n}) / (a\$1) ^ Suc n"
```
```  2097
```
```  2098 definition "fps_inv a = Abs_fps (compinv a)"
```
```  2099
```
```  2100 lemma fps_inv: assumes a0: "a\$0 = 0" and a1: "a\$1 \<noteq> 0"
```
```  2101   shows "fps_inv a oo a = X"
```
```  2102 proof-
```
```  2103   let ?i = "fps_inv a oo a"
```
```  2104   {fix n
```
```  2105     have "?i \$n = X\$n"
```
```  2106     proof(induct n rule: nat_less_induct)
```
```  2107       fix n assume h: "\<forall>m<n. ?i\$m = X\$m"
```
```  2108       {assume "n=0" hence "?i \$n = X\$n" using a0
```
```  2109           by (simp add: fps_compose_nth fps_inv_def)}
```
```  2110       moreover
```
```  2111       {fix n1 assume n1: "n = Suc n1"
```
```  2112         have "?i \$ n = setsum (\<lambda>i. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1} + fps_inv a \$ Suc n1 * (a \$ 1)^ Suc n1"
```
```  2113           by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0]
```
```  2114                    del: power_Suc)
```
```  2115         also have "\<dots> = setsum (\<lambda>i. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1} + (X\$ Suc n1 - setsum (\<lambda>i. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1})"
```
```  2116           using a0 a1 n1 by (simp add: fps_inv_def)
```
```  2117         also have "\<dots> = X\$n" using n1 by simp
```
```  2118         finally have "?i \$ n = X\$n" .}
```
```  2119       ultimately show "?i \$ n = X\$n" by (cases n, auto)
```
```  2120     qed}
```
```  2121   then show ?thesis by (simp add: fps_eq_iff)
```
```  2122 qed
```
```  2123
```
```  2124
```
```  2125 fun gcompinv :: "'a fps \<Rightarrow> 'a fps \<Rightarrow> nat \<Rightarrow> 'a::{field}" where
```
```  2126   "gcompinv b a 0 = b\$0"
```
```  2127 | "gcompinv b a (Suc n) = (b\$ Suc n - setsum (\<lambda>i. (gcompinv b a i) * (a^i)\$Suc n) {0 .. n}) / (a\$1) ^ Suc n"
```
```  2128
```
```  2129 definition "fps_ginv b a = Abs_fps (gcompinv b a)"
```
```  2130
```
```  2131 lemma fps_ginv: assumes a0: "a\$0 = 0" and a1: "a\$1 \<noteq> 0"
```
```  2132   shows "fps_ginv b a oo a = b"
```
```  2133 proof-
```
```  2134   let ?i = "fps_ginv b a oo a"
```
```  2135   {fix n
```
```  2136     have "?i \$n = b\$n"
```
```  2137     proof(induct n rule: nat_less_induct)
```
```  2138       fix n assume h: "\<forall>m<n. ?i\$m = b\$m"
```
```  2139       {assume "n=0" hence "?i \$n = b\$n" using a0
```
```  2140           by (simp add: fps_compose_nth fps_ginv_def)}
```
```  2141       moreover
```
```  2142       {fix n1 assume n1: "n = Suc n1"
```
```  2143         have "?i \$ n = setsum (\<lambda>i. (fps_ginv b a \$ i) * (a^i)\$n) {0 .. n1} + fps_ginv b a \$ Suc n1 * (a \$ 1)^ Suc n1"
```
```  2144           by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0]
```
```  2145                    del: power_Suc)
```
```  2146         also have "\<dots> = setsum (\<lambda>i. (fps_ginv b a \$ i) * (a^i)\$n) {0 .. n1} + (b\$ Suc n1 - setsum (\<lambda>i. (fps_ginv b a \$ i) * (a^i)\$n) {0 .. n1})"
```
```  2147           using a0 a1 n1 by (simp add: fps_ginv_def)
```
```  2148         also have "\<dots> = b\$n" using n1 by simp
```
```  2149         finally have "?i \$ n = b\$n" .}
```
```  2150       ultimately show "?i \$ n = b\$n" by (cases n, auto)
```
```  2151     qed}
```
```  2152   then show ?thesis by (simp add: fps_eq_iff)
```
```  2153 qed
```
```  2154
```
```  2155 lemma fps_inv_ginv: "fps_inv = fps_ginv X"
```
```  2156   apply (auto simp add: fun_eq_iff fps_eq_iff fps_inv_def fps_ginv_def)
```
```  2157   apply (induct_tac n rule: nat_less_induct, auto)
```
```  2158   apply (case_tac na)
```
```  2159   apply simp
```
```  2160   apply simp
```
```  2161   done
```
```  2162
```
```  2163 lemma fps_compose_1[simp]: "1 oo a = 1"
```
```  2164   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta)
```
```  2165
```
```  2166 lemma fps_compose_0[simp]: "0 oo a = 0"
```
```  2167   by (simp add: fps_eq_iff fps_compose_nth)
```
```  2168
```
```  2169 lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a\$0)"
```
```  2170   by (auto simp add: fps_eq_iff fps_compose_nth power_0_left setsum_0')
```
```  2171
```
```  2172 lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)"
```
```  2173   by (simp add: fps_eq_iff fps_compose_nth field_simps setsum_addf)
```
```  2174
```
```  2175 lemma fps_compose_setsum_distrib: "(setsum f S) oo a = setsum (\<lambda>i. f i oo a) S"
```
```  2176 proof-
```
```  2177   {assume "\<not> finite S" hence ?thesis by simp}
```
```  2178   moreover
```
```  2179   {assume fS: "finite S"
```
```  2180     have ?thesis
```
```  2181     proof(rule finite_induct[OF fS])
```
```  2182       show "setsum f {} oo a = (\<Sum>i\<in>{}. f i oo a)" by simp
```
```  2183     next
```
```  2184       fix x F assume fF: "finite F" and xF: "x \<notin> F" and h: "setsum f F oo a = setsum (\<lambda>i. f i oo a) F"
```
```  2185       show "setsum f (insert x F) oo a  = setsum (\<lambda>i. f i oo a) (insert x F)"
```
```  2186         using fF xF h by (simp add: fps_compose_add_distrib)
```
```  2187     qed}
```
```  2188   ultimately show ?thesis by blast
```
```  2189 qed
```
```  2190
```
```  2191 lemma convolution_eq:
```
```  2192   "setsum (%i. a (i :: nat) * b (n - i)) {0 .. n} = setsum (%(i,j). a i * b j) {(i,j). i <= n \<and> j \<le> n \<and> i + j = n}"
```
```  2193   apply (rule setsum_reindex_cong[where f=fst])
```
```  2194   apply (clarsimp simp add: inj_on_def)
```
```  2195   apply (auto simp add: set_eq_iff image_iff)
```
```  2196   apply (rule_tac x= "x" in exI)
```
```  2197   apply clarsimp
```
```  2198   apply (rule_tac x="n - x" in exI)
```
```  2199   apply arith
```
```  2200   done
```
```  2201
```
```  2202 lemma product_composition_lemma:
```
```  2203   assumes c0: "c\$0 = (0::'a::idom)" and d0: "d\$0 = 0"
```
```  2204   shows "((a oo c) * (b oo d))\$n = setsum (%(k,m). a\$k * b\$m * (c^k * d^m) \$ n) {(k,m). k + m \<le> n}" (is "?l = ?r")
```
```  2205 proof-
```
```  2206   let ?S = "{(k\<Colon>nat, m\<Colon>nat). k + m \<le> n}"
```
```  2207   have s: "?S \<subseteq> {0..n} <*> {0..n}" by (auto simp add: subset_eq)
```
```  2208   have f: "finite {(k\<Colon>nat, m\<Colon>nat). k + m \<le> n}"
```
```  2209     apply (rule finite_subset[OF s])
```
```  2210     by auto
```
```  2211   have "?r =  setsum (%i. setsum (%(k,m). a\$k * (c^k)\$i * b\$m * (d^m) \$ (n - i)) {(k,m). k + m \<le> n}) {0..n}"
```
```  2212     apply (simp add: fps_mult_nth setsum_right_distrib)
```
```  2213     apply (subst setsum_commute)
```
```  2214     apply (rule setsum_cong2)
```
```  2215     by (auto simp add: field_simps)
```
```  2216   also have "\<dots> = ?l"
```
```  2217     apply (simp add: fps_mult_nth fps_compose_nth setsum_product)
```
```  2218     apply (rule setsum_cong2)
```
```  2219     apply (simp add: setsum_cartesian_product mult_assoc)
```
```  2220     apply (rule setsum_mono_zero_right[OF f])
```
```  2221     apply (simp add: subset_eq) apply presburger
```
```  2222     apply clarsimp
```
```  2223     apply (rule ccontr)
```
```  2224     apply (clarsimp simp add: not_le)
```
```  2225     apply (case_tac "x < aa")
```
```  2226     apply simp
```
```  2227     apply (frule_tac startsby_zero_power_prefix[rule_format, OF c0])
```
```  2228     apply blast
```
```  2229     apply simp
```
```  2230     apply (frule_tac startsby_zero_power_prefix[rule_format, OF d0])
```
```  2231     apply blast
```
```  2232     done
```
```  2233   finally show ?thesis by simp
```
```  2234 qed
```
```  2235
```
```  2236 lemma product_composition_lemma':
```
```  2237   assumes c0: "c\$0 = (0::'a::idom)" and d0: "d\$0 = 0"
```
```  2238   shows "((a oo c) * (b oo d))\$n = setsum (%k. setsum (%m. a\$k * b\$m * (c^k * d^m) \$ n) {0..n}) {0..n}" (is "?l = ?r")
```
```  2239   unfolding product_composition_lemma[OF c0 d0]
```
```  2240   unfolding setsum_cartesian_product
```
```  2241   apply (rule setsum_mono_zero_left)
```
```  2242   apply simp
```
```  2243   apply (clarsimp simp add: subset_eq)
```
```  2244   apply clarsimp
```
```  2245   apply (rule ccontr)
```
```  2246   apply (subgoal_tac "(c^aa * d^ba) \$ n = 0")
```
```  2247   apply simp
```
```  2248   unfolding fps_mult_nth
```
```  2249   apply (rule setsum_0')
```
```  2250   apply (clarsimp simp add: not_le)
```
```  2251   apply (case_tac "aaa < aa")
```
```  2252   apply (rule startsby_zero_power_prefix[OF c0, rule_format])
```
```  2253   apply simp
```
```  2254   apply (subgoal_tac "n - aaa < ba")
```
```  2255   apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format])
```
```  2256   apply simp
```
```  2257   apply arith
```
```  2258   done
```
```  2259
```
```  2260
```
```  2261 lemma setsum_pair_less_iff:
```
```  2262   "setsum (%((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m \<le> n} = setsum (%s. setsum (%i. a i * b (s - i) * c s) {0..s}) {0..n}" (is "?l = ?r")
```
```  2263 proof-
```
```  2264   let ?KM=  "{(k,m). k + m \<le> n}"
```
```  2265   let ?f = "%s. UNION {(0::nat)..s} (%i. {(i,s - i)})"
```
```  2266   have th0: "?KM = UNION {0..n} ?f"
```
```  2267     apply (simp add: set_eq_iff)
```
```  2268     apply arith (* FIXME: VERY slow! *)
```
```  2269     done
```
```  2270   show "?l = ?r "
```
```  2271     unfolding th0
```
```  2272     apply (subst setsum_UN_disjoint)
```
```  2273     apply auto
```
```  2274     apply (subst setsum_UN_disjoint)
```
```  2275     apply auto
```
```  2276     done
```
```  2277 qed
```
```  2278
```
```  2279 lemma fps_compose_mult_distrib_lemma:
```
```  2280   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2281   shows "((a oo c) * (b oo c))\$n = setsum (%s. setsum (%i. a\$i * b\$(s - i) * (c^s) \$ n) {0..s}) {0..n}" (is "?l = ?r")
```
```  2282   unfolding product_composition_lemma[OF c0 c0] power_add[symmetric]
```
```  2283   unfolding setsum_pair_less_iff[where a = "%k. a\$k" and b="%m. b\$m" and c="%s. (c ^ s)\$n" and n = n] ..
```
```  2284
```
```  2285
```
```  2286 lemma fps_compose_mult_distrib:
```
```  2287   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2288   shows "(a * b) oo c = (a oo c) * (b oo c)" (is "?l = ?r")
```
```  2289   apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma[OF c0])
```
```  2290   by (simp add: fps_compose_nth fps_mult_nth setsum_left_distrib)
```
```  2291 lemma fps_compose_setprod_distrib:
```
```  2292   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2293   shows "(setprod a S) oo c = setprod (%k. a k oo c) S" (is "?l = ?r")
```
```  2294   apply (cases "finite S")
```
```  2295   apply simp_all
```
```  2296   apply (induct S rule: finite_induct)
```
```  2297   apply simp
```
```  2298   apply (simp add: fps_compose_mult_distrib[OF c0])
```
```  2299   done
```
```  2300
```
```  2301 lemma fps_compose_power:   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2302   shows "(a oo c)^n = a^n oo c" (is "?l = ?r")
```
```  2303 proof-
```
```  2304   {assume "n=0" then have ?thesis by simp}
```
```  2305   moreover
```
```  2306   {fix m assume m: "n = Suc m"
```
```  2307     have th0: "a^n = setprod (%k. a) {0..m}" "(a oo c) ^ n = setprod (%k. a oo c) {0..m}"
```
```  2308       by (simp_all add: setprod_constant m)
```
```  2309     then have ?thesis
```
```  2310       by (simp add: fps_compose_setprod_distrib[OF c0])}
```
```  2311   ultimately show ?thesis by (cases n, auto)
```
```  2312 qed
```
```  2313
```
```  2314 lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)"
```
```  2315   by (simp add: fps_eq_iff fps_compose_nth field_simps setsum_negf[symmetric])
```
```  2316
```
```  2317 lemma fps_compose_sub_distrib:
```
```  2318   shows "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)"
```
```  2319   unfolding diff_minus fps_compose_uminus fps_compose_add_distrib ..
```
```  2320
```
```  2321 lemma X_fps_compose:"X oo a = Abs_fps (\<lambda>n. if n = 0 then (0::'a::comm_ring_1) else a\$n)"
```
```  2322   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta power_Suc)
```
```  2323
```
```  2324 lemma fps_inverse_compose:
```
```  2325   assumes b0: "(b\$0 :: 'a::field) = 0" and a0: "a\$0 \<noteq> 0"
```
```  2326   shows "inverse a oo b = inverse (a oo b)"
```
```  2327 proof-
```
```  2328   let ?ia = "inverse a"
```
```  2329   let ?ab = "a oo b"
```
```  2330   let ?iab = "inverse ?ab"
```
```  2331
```
```  2332 from a0 have ia0: "?ia \$ 0 \<noteq> 0" by (simp )
```
```  2333 from a0 have ab0: "?ab \$ 0 \<noteq> 0" by (simp add: fps_compose_def)
```
```  2334 have "(?ia oo b) *  (a oo b) = 1"
```
```  2335 unfolding fps_compose_mult_distrib[OF b0, symmetric]
```
```  2336 unfolding inverse_mult_eq_1[OF a0]
```
```  2337 fps_compose_1 ..
```
```  2338
```
```  2339 then have "(?ia oo b) *  (a oo b) * ?iab  = 1 * ?iab" by simp
```
```  2340 then have "(?ia oo b) *  (?iab * (a oo b))  = ?iab" by simp
```
```  2341 then show ?thesis
```
```  2342   unfolding inverse_mult_eq_1[OF ab0] by simp
```
```  2343 qed
```
```  2344
```
```  2345 lemma fps_divide_compose:
```
```  2346   assumes c0: "(c\$0 :: 'a::field) = 0" and b0: "b\$0 \<noteq> 0"
```
```  2347   shows "(a/b) oo c = (a oo c) / (b oo c)"
```
```  2348     unfolding fps_divide_def fps_compose_mult_distrib[OF c0]
```
```  2349     fps_inverse_compose[OF c0 b0] ..
```
```  2350
```
```  2351 lemma gp: assumes a0: "a\$0 = (0::'a::field)"
```
```  2352   shows "(Abs_fps (\<lambda>n. 1)) oo a = 1/(1 - a)" (is "?one oo a = _")
```
```  2353 proof-
```
```  2354   have o0: "?one \$ 0 \<noteq> 0" by simp
```
```  2355   have th0: "(1 - X) \$ 0 \<noteq> (0::'a)" by simp
```
```  2356   from fps_inverse_gp[where ?'a = 'a]
```
```  2357   have "inverse ?one = 1 - X" by (simp add: fps_eq_iff)
```
```  2358   hence "inverse (inverse ?one) = inverse (1 - X)" by simp
```
```  2359   hence th: "?one = 1/(1 - X)" unfolding fps_inverse_idempotent[OF o0]
```
```  2360     by (simp add: fps_divide_def)
```
```  2361   show ?thesis unfolding th
```
```  2362     unfolding fps_divide_compose[OF a0 th0]
```
```  2363     fps_compose_1 fps_compose_sub_distrib X_fps_compose_startby0[OF a0] ..
```
```  2364 qed
```
```  2365
```
```  2366 lemma fps_const_power[simp]: "fps_const (c::'a::ring_1) ^ n = fps_const (c^n)"
```
```  2367 by (induct n, auto)
```
```  2368
```
```  2369 lemma fps_compose_radical:
```
```  2370   assumes b0: "b\$0 = (0::'a::field_char_0)"
```
```  2371   and ra0: "r (Suc k) (a\$0) ^ Suc k = a\$0"
```
```  2372   and a0: "a\$0 \<noteq> 0"
```
```  2373   shows "fps_radical r (Suc k)  a oo b = fps_radical r (Suc k) (a oo b)"
```
```  2374 proof-
```
```  2375   let ?r = "fps_radical r (Suc k)"
```
```  2376   let ?ab = "a oo b"
```
```  2377   have ab0: "?ab \$ 0 = a\$0" by (simp add: fps_compose_def)
```
```  2378   from ab0 a0 ra0 have rab0: "?ab \$ 0 \<noteq> 0" "r (Suc k) (?ab \$ 0) ^ Suc k = ?ab \$ 0" by simp_all
```
```  2379   have th00: "r (Suc k) ((a oo b) \$ 0) = (fps_radical r (Suc k) a oo b) \$ 0"
```
```  2380     by (simp add: ab0 fps_compose_def)
```
```  2381   have th0: "(?r a oo b) ^ (Suc k) = a  oo b"
```
```  2382     unfolding fps_compose_power[OF b0]
```
```  2383     unfolding iffD1[OF power_radical[of a r k], OF a0 ra0]  ..
```
```  2384   from iffD1[OF radical_unique[where r=r and k=k and b= ?ab and a = "?r a oo b", OF rab0(2) th00 rab0(1)], OF th0] show ?thesis  .
```
```  2385 qed
```
```  2386
```
```  2387 lemma fps_const_mult_apply_left:
```
```  2388   "fps_const c * (a oo b) = (fps_const c * a) oo b"
```
```  2389   by (simp add: fps_eq_iff fps_compose_nth setsum_right_distrib mult_assoc)
```
```  2390
```
```  2391 lemma fps_const_mult_apply_right:
```
```  2392   "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
```
```  2393   by (auto simp add: fps_const_mult_apply_left mult_commute)
```
```  2394
```
```  2395 lemma fps_compose_assoc:
```
```  2396   assumes c0: "c\$0 = (0::'a::idom)" and b0: "b\$0 = 0"
```
```  2397   shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
```
```  2398 proof-
```
```  2399   {fix n
```
```  2400     have "?l\$n = (setsum (\<lambda>i. (fps_const (a\$i) * b^i) oo c) {0..n})\$n"
```
```  2401       by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left setsum_right_distrib mult_assoc fps_setsum_nth)
```
```  2402     also have "\<dots> = ((setsum (\<lambda>i. fps_const (a\$i) * b^i) {0..n}) oo c)\$n"
```
```  2403       by (simp add: fps_compose_setsum_distrib)
```
```  2404     also have "\<dots> = ?r\$n"
```
```  2405       apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult_assoc)
```
```  2406       apply (rule setsum_cong2)
```
```  2407       apply (rule setsum_mono_zero_right)
```
```  2408       apply (auto simp add: not_le)
```
```  2409       by (erule startsby_zero_power_prefix[OF b0, rule_format])
```
```  2410     finally have "?l\$n = ?r\$n" .}
```
```  2411   then show ?thesis by (simp add: fps_eq_iff)
```
```  2412 qed
```
```  2413
```
```  2414
```
```  2415 lemma fps_X_power_compose:
```
```  2416   assumes a0: "a\$0=0" shows "X^k oo a = (a::('a::idom fps))^k" (is "?l = ?r")
```
```  2417 proof-
```
```  2418   {assume "k=0" hence ?thesis by simp}
```
```  2419   moreover
```
```  2420   {fix h assume h: "k = Suc h"
```
```  2421     {fix n
```
```  2422       {assume kn: "k>n" hence "?l \$ n = ?r \$n" using a0 startsby_zero_power_prefix[OF a0] h
```
```  2423           by (simp add: fps_compose_nth del: power_Suc)}
```
```  2424       moreover
```
```  2425       {assume kn: "k \<le> n"
```
```  2426         hence "?l\$n = ?r\$n"
```
```  2427           by (simp add: fps_compose_nth mult_delta_left setsum_delta)}
```
```  2428       moreover have "k >n \<or> k\<le> n"  by arith
```
```  2429       ultimately have "?l\$n = ?r\$n"  by blast}
```
```  2430     then have ?thesis unfolding fps_eq_iff by blast}
```
```  2431   ultimately show ?thesis by (cases k, auto)
```
```  2432 qed
```
```  2433
```
```  2434 lemma fps_inv_right: assumes a0: "a\$0 = 0" and a1: "a\$1 \<noteq> 0"
```
```  2435   shows "a oo fps_inv a = X"
```
```  2436 proof-
```
```  2437   let ?ia = "fps_inv a"
```
```  2438   let ?iaa = "a oo fps_inv a"
```
```  2439   have th0: "?ia \$ 0 = 0" by (simp add: fps_inv_def)
```
```  2440   have th1: "?iaa \$ 0 = 0" using a0 a1
```
```  2441     by (simp add: fps_inv_def fps_compose_nth)
```
```  2442   have th2: "X\$0 = 0" by simp
```
```  2443   from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo X" by simp
```
```  2444   then have "(a oo fps_inv a) oo a = X oo a"
```
```  2445     by (simp add: fps_compose_assoc[OF a0 th0] X_fps_compose_startby0[OF a0])
```
```  2446   with fps_compose_inj_right[OF a0 a1]
```
```  2447   show ?thesis by simp
```
```  2448 qed
```
```  2449
```
```  2450 lemma fps_inv_deriv:
```
```  2451   assumes a0:"a\$0 = (0::'a::{field})" and a1: "a\$1 \<noteq> 0"
```
```  2452   shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)"
```
```  2453 proof-
```
```  2454   let ?ia = "fps_inv a"
```
```  2455   let ?d = "fps_deriv a oo ?ia"
```
```  2456   let ?dia = "fps_deriv ?ia"
```
```  2457   have ia0: "?ia\$0 = 0" by (simp add: fps_inv_def)
```
```  2458   have th0: "?d\$0 \<noteq> 0" using a1 by (simp add: fps_compose_nth fps_deriv_nth)
```
```  2459   from fps_inv_right[OF a0 a1] have "?d * ?dia = 1"
```
```  2460     by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] )
```
```  2461   hence "inverse ?d * ?d * ?dia = inverse ?d * 1" by simp
```
```  2462   with inverse_mult_eq_1[OF th0]
```
```  2463   show "?dia = inverse ?d" by simp
```
```  2464 qed
```
```  2465
```
```  2466 lemma fps_inv_idempotent:
```
```  2467   assumes a0: "a\$0 = 0" and a1: "a\$1 \<noteq> 0"
```
```  2468   shows "fps_inv (fps_inv a) = a"
```
```  2469 proof-
```
```  2470   let ?r = "fps_inv"
```
```  2471   have ra0: "?r a \$ 0 = 0" by (simp add: fps_inv_def)
```
```  2472   from a1 have ra1: "?r a \$ 1 \<noteq> 0" by (simp add: fps_inv_def field_simps)
```
```  2473   have X0: "X\$0 = 0" by simp
```
```  2474   from fps_inv[OF ra0 ra1] have "?r (?r a) oo ?r a = X" .
```
```  2475   then have "?r (?r a) oo ?r a oo a = X oo a" by simp
```
```  2476   then have "?r (?r a) oo (?r a oo a) = a"
```
```  2477     unfolding X_fps_compose_startby0[OF a0]
```
```  2478     unfolding fps_compose_assoc[OF a0 ra0, symmetric] .
```
```  2479   then show ?thesis unfolding fps_inv[OF a0 a1] by simp
```
```  2480 qed
```
```  2481
```
```  2482 lemma fps_ginv_ginv:
```
```  2483   assumes a0: "a\$0 = 0" and a1: "a\$1 \<noteq> 0"
```
```  2484   and c0: "c\$0 = 0" and  c1: "c\$1 \<noteq> 0"
```
```  2485   shows "fps_ginv b (fps_ginv c a) = b oo a oo fps_inv c"
```
```  2486 proof-
```
```  2487   let ?r = "fps_ginv"
```
```  2488   from c0 have rca0: "?r c a \$0 = 0" by (simp add: fps_ginv_def)
```
```  2489   from a1 c1 have rca1: "?r c a \$ 1 \<noteq> 0" by (simp add: fps_ginv_def field_simps)
```
```  2490   from fps_ginv[OF rca0 rca1]
```
```  2491   have "?r b (?r c a) oo ?r c a = b" .
```
```  2492   then have "?r b (?r c a) oo ?r c a oo a = b oo a" by simp
```
```  2493   then have "?r b (?r c a) oo (?r c a oo a) = b oo a"
```
```  2494     apply (subst fps_compose_assoc)
```
```  2495     using a0 c0 by (auto simp add: fps_ginv_def)
```
```  2496   then have "?r b (?r c a) oo c = b oo a"
```
```  2497     unfolding fps_ginv[OF a0 a1] .
```
```  2498   then have "?r b (?r c a) oo c oo fps_inv c= b oo a oo fps_inv c" by simp
```
```  2499   then have "?r b (?r c a) oo (c oo fps_inv c) = b oo a oo fps_inv c"
```
```  2500     apply (subst fps_compose_assoc)
```
```  2501     using a0 c0 by (auto simp add: fps_inv_def)
```
```  2502   then show ?thesis unfolding fps_inv_right[OF c0 c1] by simp
```
```  2503 qed
```
```  2504
```
```  2505 lemma fps_ginv_deriv:
```
```  2506   assumes a0:"a\$0 = (0::'a::{field})" and a1: "a\$1 \<noteq> 0"
```
```  2507   shows "fps_deriv (fps_ginv b a) = (fps_deriv b / fps_deriv a) oo fps_ginv X a"
```
```  2508 proof-
```
```  2509   let ?ia = "fps_ginv b a"
```
```  2510   let ?iXa = "fps_ginv X a"
```
```  2511   let ?d = "fps_deriv"
```
```  2512   let ?dia = "?d ?ia"
```
```  2513   have iXa0: "?iXa \$ 0 = 0" by (simp add: fps_ginv_def)
```
```  2514   have da0: "?d a \$ 0 \<noteq> 0" using a1 by simp
```
```  2515   from fps_ginv[OF a0 a1, of b] have "?d (?ia oo a) = fps_deriv b" by simp
```
```  2516   then have "(?d ?ia oo a) * ?d a = ?d b" unfolding fps_compose_deriv[OF a0] .
```
```  2517   then have "(?d ?ia oo a) * ?d a * inverse (?d a) = ?d b * inverse (?d a)" by simp
```
```  2518   then have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a"
```
```  2519     by (simp add: fps_divide_def)
```
```  2520   then have "(?d ?ia oo a) oo ?iXa =  (?d b / ?d a) oo ?iXa "
```
```  2521     unfolding inverse_mult_eq_1[OF da0] by simp
```
```  2522   then have "?d ?ia oo (a oo ?iXa) =  (?d b / ?d a) oo ?iXa"
```
```  2523     unfolding fps_compose_assoc[OF iXa0 a0] .
```
```  2524   then show ?thesis unfolding fps_inv_ginv[symmetric]
```
```  2525     unfolding fps_inv_right[OF a0 a1] by simp
```
```  2526 qed
```
```  2527
```
```  2528 subsection{* Elementary series *}
```
```  2529
```
```  2530 subsubsection{* Exponential series *}
```
```  2531 definition "E x = Abs_fps (\<lambda>n. x^n / of_nat (fact n))"
```
```  2532
```
```  2533 lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::field_char_0) * E a" (is "?l = ?r")
```
```  2534 proof-
```
```  2535   {fix n
```
```  2536     have "?l\$n = ?r \$ n"
```
```  2537   apply (auto simp add: E_def field_simps power_Suc[symmetric]simp del: fact_Suc of_nat_Suc power_Suc)
```
```  2538   by (simp add: of_nat_mult field_simps)}
```
```  2539 then show ?thesis by (simp add: fps_eq_iff)
```
```  2540 qed
```
```  2541
```
```  2542 lemma E_unique_ODE:
```
```  2543   "fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a\$0) * E (c :: 'a::field_char_0)"
```
```  2544   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2545 proof-
```
```  2546   {assume d: ?lhs
```
```  2547   from d have th: "\<And>n. a \$ Suc n = c * a\$n / of_nat (Suc n)"
```
```  2548     by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
```
```  2549   {fix n have "a\$n = a\$0 * c ^ n/ (of_nat (fact n))"
```
```  2550       apply (induct n)
```
```  2551       apply simp
```
```  2552       unfolding th
```
```  2553       using fact_gt_zero_nat
```
```  2554       apply (simp add: field_simps del: of_nat_Suc fact_Suc)
```
```  2555       apply (drule sym)
```
```  2556       by (simp add: field_simps of_nat_mult power_Suc)}
```
```  2557   note th' = this
```
```  2558   have ?rhs
```
```  2559     by (auto simp add: fps_eq_iff fps_const_mult_left E_def intro : th')}
```
```  2560 moreover
```
```  2561 {assume h: ?rhs
```
```  2562   have ?lhs
```
```  2563     apply (subst h)
```
```  2564     apply simp
```
```  2565     apply (simp only: h[symmetric])
```
```  2566   by simp}
```
```  2567 ultimately show ?thesis by blast
```
```  2568 qed
```
```  2569
```
```  2570 lemma E_add_mult: "E (a + b) = E (a::'a::field_char_0) * E b" (is "?l = ?r")
```
```  2571 proof-
```
```  2572   have "fps_deriv (?r) = fps_const (a+b) * ?r"
```
```  2573     by (simp add: fps_const_add[symmetric] field_simps del: fps_const_add)
```
```  2574   then have "?r = ?l" apply (simp only: E_unique_ODE)
```
```  2575     by (simp add: fps_mult_nth E_def)
```
```  2576   then show ?thesis ..
```
```  2577 qed
```
```  2578
```
```  2579 lemma E_nth[simp]: "E a \$ n = a^n / of_nat (fact n)"
```
```  2580   by (simp add: E_def)
```
```  2581
```
```  2582 lemma E0[simp]: "E (0::'a::{field}) = 1"
```
```  2583   by (simp add: fps_eq_iff power_0_left)
```
```  2584
```
```  2585 lemma E_neg: "E (- a) = inverse (E (a::'a::field_char_0))"
```
```  2586 proof-
```
```  2587   from E_add_mult[of a "- a"] have th0: "E a * E (- a) = 1"
```
```  2588     by (simp )
```
```  2589   have th1: "E a \$ 0 \<noteq> 0" by simp
```
```  2590   from fps_inverse_unique[OF th1 th0] show ?thesis by simp
```
```  2591 qed
```
```  2592
```
```  2593 lemma E_nth_deriv[simp]: "fps_nth_deriv n (E (a::'a::field_char_0)) = (fps_const a)^n * (E a)"
```
```  2594   by (induct n, auto simp add: power_Suc)
```
```  2595
```
```  2596 lemma X_compose_E[simp]: "X oo E (a::'a::{field}) = E a - 1"
```
```  2597   by (simp add: fps_eq_iff X_fps_compose)
```
```  2598
```
```  2599 lemma LE_compose:
```
```  2600   assumes a: "a\<noteq>0"
```
```  2601   shows "fps_inv (E a - 1) oo (E a - 1) = X"
```
```  2602   and "(E a - 1) oo fps_inv (E a - 1) = X"
```
```  2603 proof-
```
```  2604   let ?b = "E a - 1"
```
```  2605   have b0: "?b \$ 0 = 0" by simp
```
```  2606   have b1: "?b \$ 1 \<noteq> 0" by (simp add: a)
```
```  2607   from fps_inv[OF b0 b1] show "fps_inv (E a - 1) oo (E a - 1) = X" .
```
```  2608   from fps_inv_right[OF b0 b1] show "(E a - 1) oo fps_inv (E a - 1) = X" .
```
```  2609 qed
```
```  2610
```
```  2611
```
```  2612 lemma fps_const_inverse:
```
```  2613   "a \<noteq> 0 \<Longrightarrow> inverse (fps_const (a::'a::field)) = fps_const (inverse a)"
```
```  2614   apply (auto simp add: fps_eq_iff fps_inverse_def) by (case_tac "n", auto)
```
```  2615
```
```  2616 lemma inverse_one_plus_X:
```
```  2617   "inverse (1 + X) = Abs_fps (\<lambda>n. (- 1 ::'a::{field})^n)"
```
```  2618   (is "inverse ?l = ?r")
```
```  2619 proof-
```
```  2620   have th: "?l * ?r = 1"
```
```  2621     by (auto simp add: field_simps fps_eq_iff minus_one_power_iff)
```
```  2622   have th': "?l \$ 0 \<noteq> 0" by (simp add: )
```
```  2623   from fps_inverse_unique[OF th' th] show ?thesis .
```
```  2624 qed
```
```  2625
```
```  2626 lemma E_power_mult: "(E (c::'a::field_char_0))^n = E (of_nat n * c)"
```
```  2627   by (induct n, auto simp add: field_simps E_add_mult power_Suc)
```
```  2628
```
```  2629 lemma radical_E:
```
```  2630   assumes r: "r (Suc k) 1 = 1"
```
```  2631   shows "fps_radical r (Suc k) (E (c::'a::{field_char_0})) = E (c / of_nat (Suc k))"
```
```  2632 proof-
```
```  2633   let ?ck = "(c / of_nat (Suc k))"
```
```  2634   let ?r = "fps_radical r (Suc k)"
```
```  2635   have eq0[simp]: "?ck * of_nat (Suc k) = c" "of_nat (Suc k) * ?ck = c"
```
```  2636     by (simp_all del: of_nat_Suc)
```
```  2637   have th0: "E ?ck ^ (Suc k) = E c" unfolding E_power_mult eq0 ..
```
```  2638   have th: "r (Suc k) (E c \$0) ^ Suc k = E c \$ 0"
```
```  2639     "r (Suc k) (E c \$ 0) = E ?ck \$ 0" "E c \$ 0 \<noteq> 0" using r by simp_all
```
```  2640   from th0 radical_unique[where r=r and k=k, OF th]
```
```  2641   show ?thesis by auto
```
```  2642 qed
```
```  2643
```
```  2644 lemma Ec_E1_eq:
```
```  2645   "E (1::'a::{field_char_0}) oo (fps_const c * X) = E c"
```
```  2646   apply (auto simp add: fps_eq_iff E_def fps_compose_def power_mult_distrib)
```
```  2647   by (simp add: cond_value_iff cond_application_beta setsum_delta' cong del: if_weak_cong)
```
```  2648
```
```  2649 text{* The generalized binomial theorem as a  consequence of @{thm E_add_mult} *}
```
```  2650
```
```  2651 lemma gbinomial_theorem:
```
```  2652   "((a::'a::{field_char_0, field_inverse_zero})+b) ^ n = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
```
```  2653 proof-
```
```  2654   from E_add_mult[of a b]
```
```  2655   have "(E (a + b)) \$ n = (E a * E b)\$n" by simp
```
```  2656   then have "(a + b) ^ n = (\<Sum>i\<Colon>nat = 0\<Colon>nat..n. a ^ i * b ^ (n - i)  * (of_nat (fact n) / of_nat (fact i * fact (n - i))))"
```
```  2657     by (simp add: field_simps fps_mult_nth of_nat_mult[symmetric] setsum_right_distrib)
```
```  2658   then show ?thesis
```
```  2659     apply simp
```
```  2660     apply (rule setsum_cong2)
```
```  2661     apply simp
```
```  2662     apply (frule binomial_fact[where ?'a = 'a, symmetric])
```
```  2663     by (simp add: field_simps of_nat_mult)
```
```  2664 qed
```
```  2665
```
```  2666 text{* And the nat-form -- also available from Binomial.thy *}
```
```  2667 lemma binomial_theorem: "(a+b) ^ n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
```
```  2668   using gbinomial_theorem[of "of_nat a" "of_nat b" n]
```
```  2669   unfolding of_nat_add[symmetric] of_nat_power[symmetric] of_nat_mult[symmetric] of_nat_setsum[symmetric]
```
```  2670   by simp
```
```  2671
```
```  2672 subsubsection{* Logarithmic series *}
```
```  2673
```
```  2674 lemma Abs_fps_if_0:
```
```  2675   "Abs_fps(%n. if n=0 then (v::'a::ring_1) else f n) = fps_const v + X * Abs_fps (%n. f (Suc n))"
```
```  2676   by (auto simp add: fps_eq_iff)
```
```  2677
```
```  2678 definition L:: "'a::{field_char_0} \<Rightarrow> 'a fps" where
```
```  2679   "L c \<equiv> fps_const (1/c) * Abs_fps (\<lambda>n. if n = 0 then 0 else (- 1) ^ (n - 1) / of_nat n)"
```
```  2680
```
```  2681 lemma fps_deriv_L: "fps_deriv (L c) = fps_const (1/c) * inverse (1 + X)"
```
```  2682   unfolding inverse_one_plus_X
```
```  2683   by (simp add: L_def fps_eq_iff del: of_nat_Suc)
```
```  2684
```
```  2685 lemma L_nth: "L c \$ n = (if n=0 then 0 else 1/c * ((- 1) ^ (n - 1) / of_nat n))"
```
```  2686   by (simp add: L_def field_simps)
```
```  2687
```
```  2688 lemma L_0[simp]: "L c \$ 0 = 0" by (simp add: L_def)
```
```  2689 lemma L_E_inv:
```
```  2690   assumes a: "a\<noteq> (0::'a::{field_char_0})"
```
```  2691   shows "L a = fps_inv (E a - 1)" (is "?l = ?r")
```
```  2692 proof-
```
```  2693   let ?b = "E a - 1"
```
```  2694   have b0: "?b \$ 0 = 0" by simp
```
```  2695   have b1: "?b \$ 1 \<noteq> 0" by (simp add: a)
```
```  2696   have "fps_deriv (E a - 1) oo fps_inv (E a - 1) = (fps_const a * (E a - 1) + fps_const a) oo fps_inv (E a - 1)"
```
```  2697     by (simp add: field_simps)
```
```  2698   also have "\<dots> = fps_const a * (X + 1)" apply (simp add: fps_compose_add_distrib fps_const_mult_apply_left[symmetric] fps_inv_right[OF b0 b1])
```
```  2699     by (simp add: field_simps)
```
```  2700   finally have eq: "fps_deriv (E a - 1) oo fps_inv (E a - 1) = fps_const a * (X + 1)" .
```
```  2701   from fps_inv_deriv[OF b0 b1, unfolded eq]
```
```  2702   have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (X + 1)"
```
```  2703     using a
```
```  2704     by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
```
```  2705   hence "fps_deriv ?l = fps_deriv ?r"
```
```  2706     by (simp add: fps_deriv_L add_commute fps_divide_def divide_inverse)
```
```  2707   then show ?thesis unfolding fps_deriv_eq_iff
```
```  2708     by (simp add: L_nth fps_inv_def)
```
```  2709 qed
```
```  2710
```
```  2711 lemma L_mult_add:
```
```  2712   assumes c0: "c\<noteq>0" and d0: "d\<noteq>0"
```
```  2713   shows "L c + L d = fps_const (c+d) * L (c*d)"
```
```  2714   (is "?r = ?l")
```
```  2715 proof-
```
```  2716   from c0 d0 have eq: "1/c + 1/d = (c+d)/(c*d)" by (simp add: field_simps)
```
```  2717   have "fps_deriv ?r = fps_const (1/c + 1/d) * inverse (1 + X)"
```
```  2718     by (simp add: fps_deriv_L fps_const_add[symmetric] algebra_simps del: fps_const_add)
```
```  2719   also have "\<dots> = fps_deriv ?l"
```
```  2720     apply (simp add: fps_deriv_L)
```
```  2721     by (simp add: fps_eq_iff eq)
```
```  2722   finally show ?thesis
```
```  2723     unfolding fps_deriv_eq_iff by simp
```
```  2724 qed
```
```  2725
```
```  2726 subsubsection{* Binomial series *}
```
```  2727
```
```  2728 definition "fps_binomial a = Abs_fps (\<lambda>n. a gchoose n)"
```
```  2729
```
```  2730 lemma fps_binomial_nth[simp]: "fps_binomial a \$ n = a gchoose n"
```
```  2731   by (simp add: fps_binomial_def)
```
```  2732
```
```  2733 lemma fps_binomial_ODE_unique:
```
```  2734   fixes c :: "'a::field_char_0"
```
```  2735   shows "fps_deriv a = (fps_const c * a) / (1 + X) \<longleftrightarrow> a = fps_const (a\$0) * fps_binomial c"
```
```  2736   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2737 proof-
```
```  2738   let ?da = "fps_deriv a"
```
```  2739   let ?x1 = "(1 + X):: 'a fps"
```
```  2740   let ?l = "?x1 * ?da"
```
```  2741   let ?r = "fps_const c * a"
```
```  2742   have x10: "?x1 \$ 0 \<noteq> 0" by simp
```
```  2743   have "?l = ?r \<longleftrightarrow> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
```
```  2744   also have "\<dots> \<longleftrightarrow> ?da = (fps_const c * a) / ?x1"
```
```  2745     apply (simp only: fps_divide_def  mult_assoc[symmetric] inverse_mult_eq_1[OF x10])
```
```  2746     by (simp add: field_simps)
```
```  2747   finally have eq: "?l = ?r \<longleftrightarrow> ?lhs" by simp
```
```  2748   moreover
```
```  2749   {assume h: "?l = ?r"
```
```  2750     {fix n
```
```  2751       from h have lrn: "?l \$ n = ?r\$n" by simp
```
```  2752
```
```  2753       from lrn
```
```  2754       have "a\$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a \$n"
```
```  2755         apply (simp add: field_simps del: of_nat_Suc)
```
```  2756         by (cases n, simp_all add: field_simps del: of_nat_Suc)
```
```  2757     }
```
```  2758     note th0 = this
```
```  2759     {fix n have "a\$n = (c gchoose n) * a\$0"
```
```  2760       proof(induct n)
```
```  2761         case 0 thus ?case by simp
```
```  2762       next
```
```  2763         case (Suc m)
```
```  2764         thus ?case unfolding th0
```
```  2765           apply (simp add: field_simps del: of_nat_Suc)
```
```  2766           unfolding mult_assoc[symmetric] gbinomial_mult_1
```
```  2767           by (simp add: field_simps)
```
```  2768       qed}
```
```  2769     note th1 = this
```
```  2770     have ?rhs
```
```  2771       apply (simp add: fps_eq_iff)
```
```  2772       apply (subst th1)
```
```  2773       by (simp add: field_simps)}
```
```  2774   moreover
```
```  2775   {assume h: ?rhs
```
```  2776   have th00:"\<And>x y. x * (a\$0 * y) = a\$0 * (x*y)" by (simp add: mult_commute)
```
```  2777     have "?l = ?r"
```
```  2778       apply (subst h)
```
```  2779       apply (subst (2) h)
```
```  2780       apply (clarsimp simp add: fps_eq_iff field_simps)
```
```  2781       unfolding mult_assoc[symmetric] th00 gbinomial_mult_1
```
```  2782       by (simp add: field_simps gbinomial_mult_1)}
```
```  2783   ultimately show ?thesis by blast
```
```  2784 qed
```
```  2785
```
```  2786 lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + X)"
```
```  2787 proof-
```
```  2788   let ?a = "fps_binomial c"
```
```  2789   have th0: "?a = fps_const (?a\$0) * ?a" by (simp)
```
```  2790   from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis .
```
```  2791 qed
```
```  2792
```
```  2793 lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r")
```
```  2794 proof-
```
```  2795   let ?P = "?r - ?l"
```
```  2796   let ?b = "fps_binomial"
```
```  2797   let ?db = "\<lambda>x. fps_deriv (?b x)"
```
```  2798   have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)"  by simp
```
```  2799   also have "\<dots> = inverse (1 + X) * (fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))"
```
```  2800     unfolding fps_binomial_deriv
```
```  2801     by (simp add: fps_divide_def field_simps)
```
```  2802   also have "\<dots> = (fps_const (c + d)/ (1 + X)) * ?P"
```
```  2803     by (simp add: field_simps fps_divide_def fps_const_add[symmetric] del: fps_const_add)
```
```  2804   finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + X)"
```
```  2805     by (simp add: fps_divide_def)
```
```  2806   have "?P = fps_const (?P\$0) * ?b (c + d)"
```
```  2807     unfolding fps_binomial_ODE_unique[symmetric]
```
```  2808     using th0 by simp
```
```  2809   hence "?P = 0" by (simp add: fps_mult_nth)
```
```  2810   then show ?thesis by simp
```
```  2811 qed
```
```  2812
```
```  2813 lemma fps_minomial_minus_one: "fps_binomial (- 1) = inverse (1 + X)"
```
```  2814   (is "?l = inverse ?r")
```
```  2815 proof-
```
```  2816   have th: "?r\$0 \<noteq> 0" by simp
```
```  2817   have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + X)"
```
```  2818     by (simp add: fps_inverse_deriv[OF th] fps_divide_def power2_eq_square mult_commute fps_const_neg[symmetric] del: fps_const_neg)
```
```  2819   have eq: "inverse ?r \$ 0 = 1"
```
```  2820     by (simp add: fps_inverse_def)
```
```  2821   from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + X)" "- 1"] th'] eq
```
```  2822   show ?thesis by (simp add: fps_inverse_def)
```
```  2823 qed
```
```  2824
```
```  2825 text{* Vandermonde's Identity as a consequence *}
```
```  2826 lemma gbinomial_Vandermonde: "setsum (\<lambda>k. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
```
```  2827 proof-
```
```  2828   let ?ba = "fps_binomial a"
```
```  2829   let ?bb = "fps_binomial b"
```
```  2830   let ?bab = "fps_binomial (a + b)"
```
```  2831   from fps_binomial_add_mult[of a b] have "?bab \$ n = (?ba * ?bb)\$n" by simp
```
```  2832   then show ?thesis by (simp add: fps_mult_nth)
```
```  2833 qed
```
```  2834
```
```  2835 lemma binomial_Vandermonde: "setsum (\<lambda>k. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
```
```  2836   using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n]
```
```  2837
```
```  2838   apply (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric] of_nat_setsum[symmetric] of_nat_add[symmetric])
```
```  2839   by simp
```
```  2840
```
```  2841 lemma binomial_Vandermonde_same: "setsum (\<lambda>k. (n choose k)^2) {0..n} = (2*n) choose n"
```
```  2842   using binomial_Vandermonde[of n n n,symmetric]
```
```  2843   unfolding nat_mult_2 apply (simp add: power2_eq_square)
```
```  2844   apply (rule setsum_cong2)
```
```  2845   by (auto intro:  binomial_symmetric)
```
```  2846
```
```  2847 lemma Vandermonde_pochhammer_lemma:
```
```  2848   fixes a :: "'a::field_char_0"
```
```  2849   assumes b: "\<forall> j\<in>{0 ..<n}. b \<noteq> of_nat j"
```
```  2850   shows "setsum (%k. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) / (of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} = pochhammer (- (a+ b)) n / pochhammer (- b) n" (is "?l = ?r")
```
```  2851 proof-
```
```  2852   let ?m1 = "%m. (- 1 :: 'a) ^ m"
```
```  2853   let ?f = "%m. of_nat (fact m)"
```
```  2854   let ?p = "%(x::'a). pochhammer (- x)"
```
```  2855   from b have bn0: "?p b n \<noteq> 0" unfolding pochhammer_eq_0_iff by simp
```
```  2856   {fix k assume kn: "k \<in> {0..n}"
```
```  2857     {assume c:"pochhammer (b - of_nat n + 1) n = 0"
```
```  2858       then obtain j where j: "j < n" "b - of_nat n + 1 = - of_nat j"
```
```  2859         unfolding pochhammer_eq_0_iff by blast
```
```  2860       from j have "b = of_nat n - of_nat j - of_nat 1"
```
```  2861         by (simp add: algebra_simps)
```
```  2862       then have "b = of_nat (n - j - 1)"
```
```  2863         using j kn by (simp add: of_nat_diff)
```
```  2864       with b have False using j by auto}
```
```  2865     then have nz: "pochhammer (1 + b - of_nat n) n \<noteq> 0"
```
```  2866       by (auto simp add: algebra_simps)
```
```  2867
```
```  2868     from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k \<noteq> 0"
```
```  2869       by (rule pochhammer_neq_0_mono)
```
```  2870     {assume k0: "k = 0 \<or> n =0"
```
```  2871       then have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
```
```  2872         using kn
```
```  2873         by (cases "k=0", simp_all add: gbinomial_pochhammer)}
```
```  2874     moreover
```
```  2875     {assume n0: "n \<noteq> 0" and k0: "k \<noteq> 0"
```
```  2876       then obtain m where m: "n = Suc m" by (cases n, auto)
```
```  2877       from k0 obtain h where h: "k = Suc h" by (cases k, auto)
```
```  2878       {assume kn: "k = n"
```
```  2879         then have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
```
```  2880           using kn pochhammer_minus'[where k=k and n=n and b=b]
```
```  2881           apply (simp add:  pochhammer_same)
```
```  2882           using bn0
```
```  2883           by (simp add: field_simps power_add[symmetric])}
```
```  2884       moreover
```
```  2885       {assume nk: "k \<noteq> n"
```
```  2886         have m1nk: "?m1 n = setprod (%i. - 1) {0..m}"
```
```  2887           "?m1 k = setprod (%i. - 1) {0..h}"
```
```  2888           by (simp_all add: setprod_constant m h)
```
```  2889         from kn nk have kn': "k < n" by simp
```
```  2890         have bnz0: "pochhammer (b - of_nat n + 1) k \<noteq> 0"
```
```  2891           using bn0 kn
```
```  2892           unfolding pochhammer_eq_0_iff
```
```  2893           apply auto
```
```  2894           apply (erule_tac x= "n - ka - 1" in allE)
```
```  2895           by (auto simp add: algebra_simps of_nat_diff)
```
```  2896         have eq1: "setprod (%k. (1::'a) + of_nat m - of_nat k) {0 .. h} = setprod of_nat {Suc (m - h) .. Suc m}"
```
```  2897           apply (rule strong_setprod_reindex_cong[where f="%k. Suc m - k "])
```
```  2898           using kn' h m
```
```  2899           apply (auto simp add: inj_on_def image_def)
```
```  2900           apply (rule_tac x="Suc m - x" in bexI)
```
```  2901           apply (simp_all add: of_nat_diff)
```
```  2902           done
```
```  2903
```
```  2904         have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
```
```  2905           unfolding m1nk
```
```  2906
```
```  2907           unfolding m h pochhammer_Suc_setprod
```
```  2908           apply (simp add: field_simps del: fact_Suc id_def)
```
```  2909           unfolding fact_altdef_nat id_def
```
```  2910           unfolding of_nat_setprod
```
```  2911           unfolding setprod_timesf[symmetric]
```
```  2912           apply auto
```
```  2913           unfolding eq1
```
```  2914           apply (subst setprod_Un_disjoint[symmetric])
```
```  2915           apply (auto)
```
```  2916           apply (rule setprod_cong)
```
```  2917           apply auto
```
```  2918           done
```
```  2919         have th20: "?m1 n * ?p b n = setprod (%i. b - of_nat i) {0..m}"
```
```  2920           unfolding m1nk
```
```  2921           unfolding m h pochhammer_Suc_setprod
```
```  2922           unfolding setprod_timesf[symmetric]
```
```  2923           apply (rule setprod_cong)
```
```  2924           apply auto
```
```  2925           done
```
```  2926         have th21:"pochhammer (b - of_nat n + 1) k = setprod (%i. b - of_nat i) {n - k .. n - 1}"
```
```  2927           unfolding h m
```
```  2928           unfolding pochhammer_Suc_setprod
```
```  2929           apply (rule strong_setprod_reindex_cong[where f="%k. n - 1 - k"])
```
```  2930           using kn
```
```  2931           apply (auto simp add: inj_on_def m h image_def)
```
```  2932           apply (rule_tac x= "m - x" in bexI)
```
```  2933           by (auto simp add: of_nat_diff)
```
```  2934
```
```  2935         have "?m1 n * ?p b n = pochhammer (b - of_nat n + 1) k * setprod (%i. b - of_nat i) {0.. n - k - 1}"
```
```  2936           unfolding th20 th21
```
```  2937           unfolding h m
```
```  2938           apply (subst setprod_Un_disjoint[symmetric])
```
```  2939           using kn' h m
```
```  2940           apply auto
```
```  2941           apply (rule setprod_cong)
```
```  2942           apply auto
```
```  2943           done
```
```  2944         then have th2: "(?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k = setprod (%i. b - of_nat i) {0.. n - k - 1}"
```
```  2945           using nz' by (simp add: field_simps)
```
```  2946         have "(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k) = ((?m1 k * ?p (of_nat n) k) / ?f n) * ((?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k)"
```
```  2947           using bnz0
```
```  2948           by (simp add: field_simps)
```
```  2949         also have "\<dots> = b gchoose (n - k)"
```
```  2950           unfolding th1 th2
```
```  2951           using kn' by (simp add: gbinomial_def)
```
```  2952         finally have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" by simp}
```
```  2953       ultimately have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
```
```  2954         by (cases "k =n", auto)}
```
```  2955     ultimately have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" "pochhammer (1 + b - of_nat n) k \<noteq> 0 "
```
```  2956       using nz'
```
```  2957       apply (cases "n=0", auto)
```
```  2958       by (cases "k", auto)}
```
```  2959   note th00 = this
```
```  2960   have "?r = ((a + b) gchoose n) * (of_nat (fact n)/ (?m1 n * pochhammer (- b) n))"
```
```  2961     unfolding gbinomial_pochhammer
```
```  2962     using bn0 by (auto simp add: field_simps)
```
```  2963   also have "\<dots> = ?l"
```
```  2964     unfolding gbinomial_Vandermonde[symmetric]
```
```  2965     apply (simp add: th00)
```
```  2966     unfolding gbinomial_pochhammer
```
```  2967     using bn0 apply (simp add: setsum_left_distrib setsum_right_distrib field_simps)
```
```  2968     apply (rule setsum_cong2)
```
```  2969     apply (drule th00(2))
```
```  2970     by (simp add: field_simps power_add[symmetric])
```
```  2971   finally show ?thesis by simp
```
```  2972 qed
```
```  2973
```
```  2974
```
```  2975 lemma Vandermonde_pochhammer:
```
```  2976    fixes a :: "'a::field_char_0"
```
```  2977   assumes c: "ALL i : {0..< n}. c \<noteq> - of_nat i"
```
```  2978   shows "setsum (%k. (pochhammer a k * pochhammer (- (of_nat n)) k) / (of_nat (fact k) * pochhammer c k)) {0..n} = pochhammer (c - a) n / pochhammer c n"
```
```  2979 proof-
```
```  2980   let ?a = "- a"
```
```  2981   let ?b = "c + of_nat n - 1"
```
```  2982   have h: "\<forall> j \<in>{0..< n}. ?b \<noteq> of_nat j" using c
```
```  2983     apply (auto simp add: algebra_simps of_nat_diff)
```
```  2984     apply (erule_tac x= "n - j - 1" in ballE)
```
```  2985     by (auto simp add: of_nat_diff algebra_simps)
```
```  2986   have th0: "pochhammer (- (?a + ?b)) n = (- 1)^n * pochhammer (c - a) n"
```
```  2987     unfolding pochhammer_minus[OF le_refl]
```
```  2988     by (simp add: algebra_simps)
```
```  2989   have th1: "pochhammer (- ?b) n = (- 1)^n * pochhammer c n"
```
```  2990     unfolding pochhammer_minus[OF le_refl]
```
```  2991     by simp
```
```  2992   have nz: "pochhammer c n \<noteq> 0" using c
```
```  2993     by (simp add: pochhammer_eq_0_iff)
```
```  2994   from Vandermonde_pochhammer_lemma[where a = "?a" and b="?b" and n=n, OF h, unfolded th0 th1]
```
```  2995   show ?thesis using nz by (simp add: field_simps setsum_right_distrib)
```
```  2996 qed
```
```  2997
```
```  2998 subsubsection{* Formal trigonometric functions  *}
```
```  2999
```
```  3000 definition "fps_sin (c::'a::field_char_0) =
```
```  3001   Abs_fps (\<lambda>n. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"
```
```  3002
```
```  3003 definition "fps_cos (c::'a::field_char_0) =
```
```  3004   Abs_fps (\<lambda>n. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"
```
```  3005
```
```  3006 lemma fps_sin_deriv:
```
```  3007   "fps_deriv (fps_sin c) = fps_const c * fps_cos c"
```
```  3008   (is "?lhs = ?rhs")
```
```  3009 proof (rule fps_ext)
```
```  3010   fix n::nat
```
```  3011     {assume en: "even n"
```
```  3012       have "?lhs\$n = of_nat (n+1) * (fps_sin c \$ (n+1))" by simp
```
```  3013       also have "\<dots> = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))"
```
```  3014         using en by (simp add: fps_sin_def)
```
```  3015       also have "\<dots> = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
```
```  3016         unfolding fact_Suc of_nat_mult
```
```  3017         by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  3018       also have "\<dots> = (- 1)^(n div 2) *c^Suc n / of_nat (fact n)"
```
```  3019         by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  3020       finally have "?lhs \$n = ?rhs\$n" using en
```
```  3021         by (simp add: fps_cos_def field_simps power_Suc )}
```
```  3022     then show "?lhs \$ n = ?rhs \$ n"
```
```  3023       by (cases "even n", simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
```
```  3024 qed
```
```  3025
```
```  3026 lemma fps_cos_deriv:
```
```  3027   "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)"
```
```  3028   (is "?lhs = ?rhs")
```
```  3029 proof (rule fps_ext)
```
```  3030   have th0: "\<And>n. - ((- 1::'a) ^ n) = (- 1)^Suc n" by (simp add: power_Suc)
```
```  3031   have th1: "\<And>n. odd n \<Longrightarrow> Suc ((n - 1) div 2) = Suc n div 2"
```
```  3032     by (case_tac n, simp_all)
```
```  3033   fix n::nat
```
```  3034     {assume en: "odd n"
```
```  3035       from en have n0: "n \<noteq>0 " by presburger
```
```  3036       have "?lhs\$n = of_nat (n+1) * (fps_cos c \$ (n+1))" by simp
```
```  3037       also have "\<dots> = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))"
```
```  3038         using en by (simp add: fps_cos_def)
```
```  3039       also have "\<dots> = (- 1)^((n + 1) div 2)*c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
```
```  3040         unfolding fact_Suc of_nat_mult
```
```  3041         by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  3042       also have "\<dots> = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)"
```
```  3043         by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  3044       also have "\<dots> = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)"
```
```  3045         unfolding th0 unfolding th1[OF en] by simp
```
```  3046       finally have "?lhs \$n = ?rhs\$n" using en
```
```  3047         by (simp add: fps_sin_def field_simps power_Suc)}
```
```  3048     then show "?lhs \$ n = ?rhs \$ n"
```
```  3049       by (cases "even n", simp_all add: fps_deriv_def fps_sin_def
```
```  3050         fps_cos_def)
```
```  3051 qed
```
```  3052
```
```  3053 lemma fps_sin_cos_sum_of_squares:
```
```  3054   "fps_cos c ^ 2 + fps_sin c ^ 2 = 1" (is "?lhs = 1")
```
```  3055 proof-
```
```  3056   have "fps_deriv ?lhs = 0"
```
```  3057     apply (simp add:  fps_deriv_power fps_sin_deriv fps_cos_deriv power_Suc)
```
```  3058     by (simp add: field_simps fps_const_neg[symmetric] del: fps_const_neg)
```
```  3059   then have "?lhs = fps_const (?lhs \$ 0)"
```
```  3060     unfolding fps_deriv_eq_0_iff .
```
```  3061   also have "\<dots> = 1"
```
```  3062     by (auto simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
```
```  3063   finally show ?thesis .
```
```  3064 qed
```
```  3065
```
```  3066 lemma divide_eq_iff: "a \<noteq> (0::'a::field) \<Longrightarrow> x / a = y \<longleftrightarrow> x = y * a"
```
```  3067 by auto
```
```  3068
```
```  3069 lemma eq_divide_iff: "a \<noteq> (0::'a::field) \<Longrightarrow> x = y / a \<longleftrightarrow> x * a = y"
```
```  3070 by auto
```
```  3071
```
```  3072 lemma fps_sin_nth_0 [simp]: "fps_sin c \$ 0 = 0"
```
```  3073 unfolding fps_sin_def by simp
```
```  3074
```
```  3075 lemma fps_sin_nth_1 [simp]: "fps_sin c \$ 1 = c"
```
```  3076 unfolding fps_sin_def by simp
```
```  3077
```
```  3078 lemma fps_sin_nth_add_2:
```
```  3079   "fps_sin c \$ (n + 2) = - (c * c * fps_sin c \$ n / (of_nat(n+1) * of_nat(n+2)))"
```
```  3080 unfolding fps_sin_def
```
```  3081 apply (cases n, simp)
```
```  3082 apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
```
```  3083 apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
```
```  3084 done
```
```  3085
```
```  3086 lemma fps_cos_nth_0 [simp]: "fps_cos c \$ 0 = 1"
```
```  3087 unfolding fps_cos_def by simp
```
```  3088
```
```  3089 lemma fps_cos_nth_1 [simp]: "fps_cos c \$ 1 = 0"
```
```  3090 unfolding fps_cos_def by simp
```
```  3091
```
```  3092 lemma fps_cos_nth_add_2:
```
```  3093   "fps_cos c \$ (n + 2) = - (c * c * fps_cos c \$ n / (of_nat(n+1) * of_nat(n+2)))"
```
```  3094 unfolding fps_cos_def
```
```  3095 apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
```
```  3096 apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
```
```  3097 done
```
```  3098
```
```  3099 lemma nat_induct2:
```
```  3100   "\<lbrakk>P 0; P 1; \<And>n. P n \<Longrightarrow> P (n + 2)\<rbrakk> \<Longrightarrow> P (n::nat)"
```
```  3101 unfolding One_nat_def numeral_2_eq_2
```
```  3102 apply (induct n rule: nat_less_induct)
```
```  3103 apply (case_tac n, simp)
```
```  3104 apply (rename_tac m, case_tac m, simp)
```
```  3105 apply (rename_tac k, case_tac k, simp_all)
```
```  3106 done
```
```  3107
```
```  3108 lemma nat_add_1_add_1: "(n::nat) + 1 + 1 = n + 2"
```
```  3109 by simp
```
```  3110
```
```  3111 lemma eq_fps_sin:
```
```  3112   assumes 0: "a \$ 0 = 0" and 1: "a \$ 1 = c"
```
```  3113   and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
```
```  3114   shows "a = fps_sin c"
```
```  3115 apply (rule fps_ext)
```
```  3116 apply (induct_tac n rule: nat_induct2)
```
```  3117 apply (simp add: fps_sin_nth_0 0)
```
```  3118 apply (simp add: fps_sin_nth_1 1 del: One_nat_def)
```
```  3119 apply (rename_tac m, cut_tac f="\<lambda>a. a \$ m" in arg_cong [OF 2])
```
```  3120 apply (simp add: nat_add_1_add_1 fps_sin_nth_add_2
```
```  3121             del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
```
```  3122 apply (subst minus_divide_left)
```
```  3123 apply (subst eq_divide_iff)
```
```  3124 apply (simp del: of_nat_add of_nat_Suc)
```
```  3125 apply (simp only: mult_ac)
```
```  3126 done
```
```  3127
```
```  3128 lemma eq_fps_cos:
```
```  3129   assumes 0: "a \$ 0 = 1" and 1: "a \$ 1 = 0"
```
```  3130   and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
```
```  3131   shows "a = fps_cos c"
```
```  3132 apply (rule fps_ext)
```
```  3133 apply (induct_tac n rule: nat_induct2)
```
```  3134 apply (simp add: fps_cos_nth_0 0)
```
```  3135 apply (simp add: fps_cos_nth_1 1 del: One_nat_def)
```
```  3136 apply (rename_tac m, cut_tac f="\<lambda>a. a \$ m" in arg_cong [OF 2])
```
```  3137 apply (simp add: nat_add_1_add_1 fps_cos_nth_add_2
```
```  3138             del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
```
```  3139 apply (subst minus_divide_left)
```
```  3140 apply (subst eq_divide_iff)
```
```  3141 apply (simp del: of_nat_add of_nat_Suc)
```
```  3142 apply (simp only: mult_ac)
```
```  3143 done
```
```  3144
```
```  3145 lemma mult_nth_0 [simp]: "(a * b) \$ 0 = a \$ 0 * b \$ 0"
```
```  3146 by (simp add: fps_mult_nth)
```
```  3147
```
```  3148 lemma mult_nth_1 [simp]: "(a * b) \$ 1 = a \$ 0 * b \$ 1 + a \$ 1 * b \$ 0"
```
```  3149 by (simp add: fps_mult_nth)
```
```  3150
```
```  3151 lemma fps_sin_add:
```
```  3152   "fps_sin (a + b) = fps_sin a * fps_cos b + fps_cos a * fps_sin b"
```
```  3153 apply (rule eq_fps_sin [symmetric], simp, simp del: One_nat_def)
```
```  3154 apply (simp del: fps_const_neg fps_const_add fps_const_mult
```
```  3155             add: fps_const_add [symmetric] fps_const_neg [symmetric]
```
```  3156                  fps_sin_deriv fps_cos_deriv algebra_simps)
```
```  3157 done
```
```  3158
```
```  3159 lemma fps_cos_add:
```
```  3160   "fps_cos (a + b) = fps_cos a * fps_cos b - fps_sin a * fps_sin b"
```
```  3161 apply (rule eq_fps_cos [symmetric], simp, simp del: One_nat_def)
```
```  3162 apply (simp del: fps_const_neg fps_const_add fps_const_mult
```
```  3163             add: fps_const_add [symmetric] fps_const_neg [symmetric]
```
```  3164                  fps_sin_deriv fps_cos_deriv algebra_simps)
```
```  3165 done
```
```  3166
```
```  3167 lemma fps_sin_even: "fps_sin (- c) = - fps_sin c"
```
```  3168   by (auto simp add: fps_eq_iff fps_sin_def)
```
```  3169
```
```  3170 lemma fps_cos_odd: "fps_cos (- c) = fps_cos c"
```
```  3171   by (auto simp add: fps_eq_iff fps_cos_def)
```
```  3172
```
```  3173 definition "fps_tan c = fps_sin c / fps_cos c"
```
```  3174
```
```  3175 lemma fps_tan_deriv: "fps_deriv(fps_tan c) = fps_const c/ (fps_cos c ^ 2)"
```
```  3176 proof-
```
```  3177   have th0: "fps_cos c \$ 0 \<noteq> 0" by (simp add: fps_cos_def)
```
```  3178   show ?thesis
```
```  3179     using fps_sin_cos_sum_of_squares[of c]
```
```  3180     apply (simp add: fps_tan_def fps_divide_deriv[OF th0] fps_sin_deriv fps_cos_deriv add: fps_const_neg[symmetric] field_simps power2_eq_square del: fps_const_neg)
```
```  3181     unfolding right_distrib[symmetric]
```
```  3182     by simp
```
```  3183 qed
```
```  3184
```
```  3185 text {* Connection to E c over the complex numbers --- Euler and De Moivre*}
```
```  3186 lemma Eii_sin_cos:
```
```  3187   "E (ii * c) = fps_cos c + fps_const ii * fps_sin c "
```
```  3188   (is "?l = ?r")
```
```  3189 proof-
```
```  3190   {fix n::nat
```
```  3191     {assume en: "even n"
```
```  3192       from en obtain m where m: "n = 2*m"
```
```  3193         unfolding even_mult_two_ex by blast
```
```  3194
```
```  3195       have "?l \$n = ?r\$n"
```
```  3196         by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
```
```  3197           power_mult power_minus)}
```
```  3198     moreover
```
```  3199     {assume on: "odd n"
```
```  3200       from on obtain m where m: "n = 2*m + 1"
```
```  3201         unfolding odd_nat_equiv_def2 by (auto simp add: nat_mult_2)
```
```  3202       have "?l \$n = ?r\$n"
```
```  3203         by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
```
```  3204           power_mult power_minus)}
```
```  3205     ultimately have "?l \$n = ?r\$n"  by blast}
```
```  3206   then show ?thesis by (simp add: fps_eq_iff)
```
```  3207 qed
```
```  3208
```
```  3209 lemma E_minus_ii_sin_cos: "E (- (ii * c)) = fps_cos c - fps_const ii * fps_sin c "
```
```  3210   unfolding minus_mult_right Eii_sin_cos by (simp add: fps_sin_even fps_cos_odd)
```
```  3211
```
```  3212 lemma fps_const_minus: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"
```
```  3213   by (simp add: fps_eq_iff fps_const_def)
```
```  3214
```
```  3215 lemma fps_number_of_fps_const: "number_of i = fps_const (number_of i :: 'a:: {comm_ring_1, number_ring})"
```
```  3216   apply (subst (2) number_of_eq)
```
```  3217 apply(rule int_induct [of _ 0])
```
```  3218 apply (simp_all add: number_of_fps_def)
```
```  3219 by (simp_all add: fps_const_add[symmetric] fps_const_minus[symmetric])
```
```  3220
```
```  3221 lemma fps_cos_Eii:
```
```  3222   "fps_cos c = (E (ii * c) + E (- ii * c)) / fps_const 2"
```
```  3223 proof-
```
```  3224   have th: "fps_cos c + fps_cos c = fps_cos c * fps_const 2"
```
```  3225     by (simp add: fps_eq_iff fps_number_of_fps_const complex_number_of_def[symmetric])
```
```  3226   show ?thesis
```
```  3227   unfolding Eii_sin_cos minus_mult_commute
```
```  3228   by (simp add: fps_sin_even fps_cos_odd fps_number_of_fps_const
```
```  3229     fps_divide_def fps_const_inverse th complex_number_of_def[symmetric])
```
```  3230 qed
```
```  3231
```
```  3232 lemma fps_sin_Eii:
```
```  3233   "fps_sin c = (E (ii * c) - E (- ii * c)) / fps_const (2*ii)"
```
```  3234 proof-
```
```  3235   have th: "fps_const \<i> * fps_sin c + fps_const \<i> * fps_sin c = fps_sin c * fps_const (2 * ii)"
```
```  3236     by (simp add: fps_eq_iff fps_number_of_fps_const complex_number_of_def[symmetric])
```
```  3237   show ?thesis
```
```  3238   unfolding Eii_sin_cos minus_mult_commute
```
```  3239   by (simp add: fps_sin_even fps_cos_odd fps_divide_def fps_const_inverse th)
```
```  3240 qed
```
```  3241
```
```  3242 lemma fps_tan_Eii:
```
```  3243   "fps_tan c = (E (ii * c) - E (- ii * c)) / (fps_const ii * (E (ii * c) + E (- ii * c)))"
```
```  3244   unfolding fps_tan_def fps_sin_Eii fps_cos_Eii mult_minus_left E_neg
```
```  3245   apply (simp add: fps_divide_def fps_inverse_mult fps_const_mult[symmetric] fps_const_inverse del: fps_const_mult)
```
```  3246   by simp
```
```  3247
```
```  3248 lemma fps_demoivre: "(fps_cos a + fps_const ii * fps_sin a)^n = fps_cos (of_nat n * a) + fps_const ii * fps_sin (of_nat n * a)"
```
```  3249   unfolding Eii_sin_cos[symmetric] E_power_mult
```
```  3250   by (simp add: mult_ac)
```
```  3251
```
```  3252 subsection {* Hypergeometric series *}
```
```  3253
```
```  3254
```
```  3255 definition "F as bs (c::'a::{field_char_0, field_inverse_zero}) = Abs_fps (%n. (foldl (%r a. r* pochhammer a n) 1 as * c^n)/ (foldl (%r b. r * pochhammer b n) 1 bs * of_nat (fact n)))"
```
```  3256
```
```  3257 lemma F_nth[simp]: "F as bs c \$ n =  (foldl (%r a. r* pochhammer a n) 1 as * c^n)/ (foldl (%r b. r * pochhammer b n) 1 bs * of_nat (fact n))"
```
```  3258   by (simp add: F_def)
```
```  3259
```
```  3260 lemma foldl_mult_start:
```
```  3261   "foldl (%r x. r * f x) (v::'a::comm_ring_1) as * x = foldl (%r x. r * f x) (v * x) as "
```
```  3262   by (induct as arbitrary: x v, auto simp add: algebra_simps)
```
```  3263
```
```  3264 lemma foldr_mult_foldl: "foldr (%x r. r * f x) as v = foldl (%r x. r * f x) (v :: 'a::comm_ring_1) as"
```
```  3265   by (induct as arbitrary: v, auto simp add: foldl_mult_start)
```
```  3266
```
```  3267 lemma F_nth_alt: "F as bs c \$ n = foldr (\<lambda>a r. r * pochhammer a n) as (c ^ n) /
```
```  3268     foldr (\<lambda>b r. r * pochhammer b n) bs (of_nat (fact n))"
```
```  3269   by (simp add: foldl_mult_start foldr_mult_foldl)
```
```  3270
```
```  3271 lemma F_E[simp]: "F [] [] c = E c"
```
```  3272   by (simp add: fps_eq_iff)
```
```  3273
```
```  3274 lemma F_1_0[simp]: "F [1] [] c = 1/(1 - fps_const c * X)"
```
```  3275 proof-
```
```  3276   let ?a = "(Abs_fps (\<lambda>n. 1)) oo (fps_const c * X)"
```
```  3277   have th0: "(fps_const c * X) \$ 0 = 0" by simp
```
```  3278   show ?thesis unfolding gp[OF th0, symmetric]
```
```  3279     by (auto simp add: fps_eq_iff pochhammer_fact[symmetric] fps_compose_nth power_mult_distrib cond_value_iff setsum_delta' cong del: if_weak_cong)
```
```  3280 qed
```
```  3281
```
```  3282 lemma F_B[simp]: "F [-a] [] (- 1) = fps_binomial a"
```
```  3283   by (simp add: fps_eq_iff gbinomial_pochhammer algebra_simps)
```
```  3284
```
```  3285 lemma F_0[simp]: "F as bs c \$0 = 1"
```
```  3286   apply simp
```
```  3287   apply (subgoal_tac "ALL as. foldl (%(r::'a) (a::'a). r) 1 as = 1")
```
```  3288   apply auto
```
```  3289   apply (induct_tac as, auto)
```
```  3290   done
```
```  3291
```
```  3292 lemma foldl_prod_prod: "foldl (%(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (%r x. r * g x) w as = foldl (%r x. r * f x * g x) (v*w) as"
```
```  3293   by (induct as arbitrary: v w, auto simp add: algebra_simps)
```
```  3294
```
```  3295
```
```  3296 lemma F_rec: "F as bs c \$ Suc n = ((foldl (%r a. r* (a + of_nat n)) c as)/ (foldl (%r b. r * (b + of_nat n)) (of_nat (Suc n)) bs )) * F as bs c \$ n"
```
```  3297   apply (simp del: of_nat_Suc of_nat_add fact_Suc)
```
```  3298   apply (simp add: foldl_mult_start del: fact_Suc of_nat_Suc)
```
```  3299   unfolding foldl_prod_prod[unfolded foldl_mult_start] pochhammer_Suc
```
```  3300   by (simp add: algebra_simps of_nat_mult)
```
```  3301
```
```  3302 lemma XD_nth[simp]: "XD a \$ n = (if n=0 then 0 else of_nat n * a\$n)"
```
```  3303   by (simp add: XD_def)
```
```  3304
```
```  3305 lemma XD_0th[simp]: "XD a \$ 0 = 0" by simp
```
```  3306 lemma XD_Suc[simp]:" XD a \$ Suc n = of_nat (Suc n) * a \$ Suc n" by simp
```
```  3307
```
```  3308 definition "XDp c a = XD a + fps_const c * a"
```
```  3309
```
```  3310 lemma XDp_nth[simp]: "XDp c a \$ n = (c + of_nat n) * a\$n"
```
```  3311   by (simp add: XDp_def algebra_simps)
```
```  3312
```
```  3313 lemma XDp_commute:
```
```  3314   shows "XDp b o XDp (c::'a::comm_ring_1) = XDp c o XDp b"
```
```  3315   by (auto simp add: XDp_def fun_eq_iff fps_eq_iff algebra_simps)
```
```  3316
```
```  3317 lemma XDp0[simp]: "XDp 0 = XD"
```
```  3318   by (simp add: fun_eq_iff fps_eq_iff)
```
```  3319
```
```  3320 lemma XDp_fps_integral[simp]:"XDp 0 (fps_integral a c) = X * a"
```
```  3321   by (simp add: fps_eq_iff fps_integral_def)
```
```  3322
```
```  3323 lemma F_minus_nat:
```
```  3324   "F [- of_nat n] [- of_nat (n + m)] (c::'a::{field_char_0, field_inverse_zero}) \$ k = (if k <= n then pochhammer (- of_nat n) k * c ^ k /
```
```  3325     (pochhammer (- of_nat (n + m)) k * of_nat (fact k)) else 0)"
```
```  3326   "F [- of_nat m] [- of_nat (m + n)] (c::'a::{field_char_0, field_inverse_zero}) \$ k = (if k <= m then pochhammer (- of_nat m) k * c ^ k /
```
```  3327     (pochhammer (- of_nat (m + n)) k * of_nat (fact k)) else 0)"
```
```  3328   by (auto simp add: pochhammer_eq_0_iff)
```
```  3329
```
```  3330 lemma setsum_eq_if: "setsum f {(n::nat) .. m} = (if m < n then 0 else f n + setsum f {n+1 .. m})"
```
```  3331   apply simp
```
```  3332   apply (subst setsum_insert[symmetric])
```
```  3333   by (auto simp add: not_less setsum_head_Suc)
```
```  3334
```
```  3335 lemma pochhammer_rec_if:
```
```  3336   "pochhammer a n = (if n = 0 then 1 else a * pochhammer (a + 1) (n - 1))"
```
```  3337   by (cases n, simp_all add: pochhammer_rec)
```
```  3338
```
```  3339 lemma XDp_foldr_nth[simp]: "foldr (%c r. XDp c o r) cs (%c. XDp c a) c0 \$ n =
```
```  3340   foldr (%c r. (c + of_nat n) * r) cs (c0 + of_nat n) * a\$n"
```
```  3341   by (induct cs arbitrary: c0, auto simp add: algebra_simps)
```
```  3342
```
```  3343 lemma genric_XDp_foldr_nth:
```
```  3344   assumes
```
```  3345   f: "ALL n c a. f c a \$ n = (of_nat n + k c) * a\$n"
```
```  3346
```
```  3347   shows "foldr (%c r. f c o r) cs (%c. g c a) c0 \$ n =
```
```  3348   foldr (%c r. (k c + of_nat n) * r) cs (g c0 a \$ n)"
```
```  3349   by (induct cs arbitrary: c0, auto simp add: algebra_simps f)
```
```  3350
```
```  3351 end
```